Chapter 3A - Rectangular Coordinate System

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 3A - Rectangular Coordinate System"

Transcription

1 - Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been practiced since ancient times, it was not until the 7th centur that geometr and algebra were joined to form the branch of mathematics called analtic geometr. French mathematician and philosopher Rene Descartes (96-60) devised a simple plan whereb two number lines were intersected at right angles with the position of a point in a plane determined b its distance from each of the lines. This sstem is called the rectangular coordinate sstem (or Cartesian coordinate sstem). -ais -ais origin (0, 0) Points are labeled with ordered pairs of real numbers,, called the coordinates of the point, which give the horizontal and vertical distance of the point from the origin, respectivel. The origin is the intersection of the -and-aes. Locations of the points in the plane are determined in relationship to this point 0,0. All points in the plane are located in one of four quadrants or on the -or-ais as illustrated below. To plot a point, start at the origin, proceed horizontall the distance and direction indicated b the -coordinate, then verticall the distance and direction indicated b the -coordinate. The resulting point is often labeled with its ordered pair coordinates and/or a capital letter. For eample, the point units to the right of the origin and units up could be labeled A,. Quadrant II Quadrant I (-, +) (+, +) (0, b) (a, 0) (0,0) Quadrant III Quadrant IV (-, -) (+, - ) Notice that the Cartesian plane has been divided into fourths. Each of these fourths is called a quadrant and the are numbered as indicated above.

2 - Chapter A Eample : Solution: Plot the following points on a rectangular coordinate sstem: A, B 0, C, D,0 E, C(-,) E(-,-) B(0,-) D(,0) A(,-) Eample : Shade the region of the coordinate plane that contains the set of ordered pairs, 0. [The set notation is read the set of all ordered pairs, such that 0.] Solution: This set describes all ordered pairs where the -coordinate is greater than 0. Plot several points that satisf the stated condition, e.g.,,, 7,,,0. These points are all located to the right of the -ais. To plot all such points we would shade all of Quadrants I and IV. We indicate that points on the -ais are not included 0 b using a dotted line. Eample : Shade the region of the coordinate plane that contains the set of ordered pairs,,. Solution: The area to the right of the dotted line designated is the set of all points where the -coordinate is greater than (shaded gra). The area between the horizontal lines designated and is the area where the -coordinate is between and (shaded red). The dark region is the intersection of these two sets of points, the set that satisfies both of the given conditions. = = = > and - < < The basis of analtic geometr lies in the connection between a set of ordered pairs and its graph on the Cartesian coordinate sstem.

3 - Chapter A Definitions: An set of ordered pairs is called a relation. The plot of ever point associated with an ordered pair in the relation is called the graph of the relation. The set of all first elements in the ordered pairs is called the domain of the relation. The set of all second elements in the ordered pairs is called the range of the relation. In Eample, we plotted five distinct points. If we consider these points as a set of ordered pairs, we have the relation,, 0,,,,,0,,. The graph is C(-,) A(,-) E(-,-) - - B(0,-) D(,0) The domain is, 0,,, and the range is,,, 0,. Infinite sets of ordered pairs can be described algebraicall and plotted (or graphed) on the coordinate sstem. Eample : Below is the graph from Eample. Recall that the graph represents all ordered pairs defined b the algebraic statement: 0. That is, the relation consists of all ordered pairs, that have an -coordinate that is a positive number. What is the domain and range of this relation? Solution: Since the relation is defined as the set of all ordered pairs where 0, the domain is 0. The -coordinates can be an real number so the range is all real numbers.

4 - Chapter A Eample : Below is the graph from Eample. What is the domain and range? = = = > and - < < Solution: The domain is all real numbers greater than. The range is all real numbers between and, including the endpoints and. Note: We often write the domain and range in interval notation. The domain for the above eample in interval notation is,. The range for the above eample in interval notation is,.

5 - Chapter A Distance Formula The marriage of algebra and geometr allows us to devise algebraic formulas to use in solving geometric problems. For eample, the formula for finding the distance between two points in the plane is derived as follows: Consider two points P, and Q,. Select a third point R, so that the three points form a right triangle with the right angle at point R. (See figure below.) Q(, ),, P( ) R( ) Note that the distance between P and R is and the distance from Q and R is. Therefore, the distance from point P to point Q, denoted d P, Q, can be found using the Pthagorean Theorem: d P, Q Because and, and because we are onl interested in positive values for the distance, we obtain the following formula. Distance Formula The distance between two points P, and Q, in the plane is d P, Q Eample : Find the distance between the points A, and B, 7. Solution: To find the distance between two points in the plane we use the distance formula d A,B. It does not matter which point ou use as, or,,so we will use the coordinates of A and B respectivel. That is,,,, and 7. Plugging into the formula, we get d A,B

6 - Chapter A Eample : triangle. Solution: Determine whether the points A,, B, and C 0, are the vertices of a right Plot and label points A, B, andc on graph paper and draw the indicated triangle. A(-,) C(0,) B(,) From inspection of the graph, point C appears to be the verte of the right triangle; therefore the line segment through A and B appears to be the hpotenuse. We will use the distance formula to find the length of each side of the triangle, and then appl the Pthagorean Theorem to determine whether the triangle is indeed a right triangle: If triangle ABC is a right triangle, d A,B d A,B d B,C 0 9 d A,C must be true. Since 7 9, triangle ABC is NOT a right triangle. Eample : Determine whether the points A, and B, are equidistant from point C,. Recall that equidistant means equal distance. That is, points A and B are equidistant from point C if and onl if the distance from A to C is equal to the distance from B to C. Algebraicall, we would write the above statement as d A,C d B,C. Solution: Plot the points on graph paper to visualize the problem B(-,-) - C(,) A(,) - Using the distance formula we can find the distance between each of the points, A and B, and point C. d A,C d B,C Because d A,C d B,C, A and B are equidistant from point C. Note that if ou plot all the points that are units from point C ou will obtain a circle.

7 - Chapter A Midpoint Formula In man instances it is important to be able to calculate the point that lies half wa between the two points on the line segment that connects them. The figure below shows points A, and B,, along with their midpoint M,. B(, ) = M(, ) Q(, ) = A(, ) P(, ) Note that Δ APM ΔBQM b ASA so that AP MQ. Solving for, we get. Similarl, Midpoint Formula. The midpoint of the line segment that connects points A, and B, is the point M, with coordinates,. The midpoint is the point M, with and. Note that the -coordinate of the midpoint is the average of the -coordinates of the the endpoints, and the -coordinate of the midpoint is the average of the -coordinates of the endpoints. The eamples below will show ou how to appl this formula. Be sure to work through them before tring the eercises. Eample : Find the midpoint of the line segment that connects the points, 9 and 7,. Solution: Plugging, 9, 7, and into the mid-point formula, we get 7 9 Therefore, the midpoint is,.

8 - Chapter A Eample : If, 8 is the midpoint of the line segment connecting A, and B, findthe coordinates of the other endpoint B. Solution: We are given the values, 8, and. We must find and. Substituting into the midpoint formula, we get and Solving the above for and, we get: Therefore, the endpoint B has coordinates, 8.

9 - Chapter A Circles The set of all points P, that are a given fied distance from a given fied point is called a circle. The fied distance, r, is called the radius and the fied point C h,k is called the center. r P(, ) C(h, k) The equation for this set of points can be found b appling the distance formula. That is, d C,P r. h k r If we square both sides of this last equation we get h k r Equation of a circle with C h, k and radius r Standard Form. h k r. Circles with center at the origin,c 0, 0 Standard Form r Eample : Find the center and radius or the circles a) 6 and b) 0. Solution: a) This circle is in the second form so its center is the origin C 0,0 and r 6 r b) This circle is in the first form so that h andk C, and r 0 r 0 Eample : Write an equation for the circle with C, and radius. Solution To write an equation for a specific circle we first write the equation for a circle in standard form: h k r, and then identif the specific values for h,k, andr.. Since we are given the center C and radius r, we can fill in values for h, k, andr as follows: h the -coordinate of C ; k the -coordinate of C ; and r. The equation of the circle is Substitute into the equation and simplif: Equation of the circle in standard form:

10 - Chapter A Eample : Find the center and radius of the circle 9. Graph the circle and find its domain and range. Solution: From the equation above, we see that h, k sothecenterisc,. Since r 9, the radius is r. Graph: Note that there are no points to the left of the point, nor to the right of,. Therefore the domain is all real numbers from to, including and, or the interval,. Similarl, the range values include all real numbers between 6 and 0, including the endpoints, which is the interval 6,0. Note:We can find the domain of the circle algebraicall b adding and subtracting the radius to the h value.,,. We can obtain the range b adding and subtracting the radius from the k value., 6,0. Eample : Write an equation for the circle with at the origin and radius. Solution: Substituting h 0, k 0, and r into the equation for a circle, we get 0 0 Since the center of the circle is the origin, we can substitute directl into the formula r to get. If we epand the equation for the circle, we get

11 - Chapter A Simplifing and arranging terms, we get We call this the general form of an equation of a circle. Notice that the coefficients of and are the same. This is our first clue that a particular equation ma be a circle. The equation h k r form is called the standard form of an equation of a circle. General Form of an Equation of a Circle c d e 0 It is important to be able to recognize that this equation also represents a circle. Note that the equation contains both an and a term and that both coeffiients equal. The general form is not as user-friendl as the standard form. We cannot find the center and radius of the circle b simpl inspecting the equation as we can with an equation in standard form. To find the center and radius of a circle that is in general form, we must reverse the above process and write the equation in standard form. For eample, 0 is the equation of a circle. (The coefficients of and are positive and equal.) Grouping the and terms and moving the constant to the other side of the equation, we get We must now complete the square on the and terms, and add the calculated amounts to both sides of the equations. Question: Answer: Wh did we add to both sides of the equation? Weaddedgobeabletowrite as a square.. So that we re not changing the equation we must either add and subtract from the same side of the equation 0, or we must add to the each side of the equation ( is equivalent to. Writing in factored form, we get the standard form of the equation: 9 From this form we can determine that the center of the circle is, and the radius is. We can use this information to graph the equation b plotting the center, and locating as man points on the circle as needed ( units in an direction from the center).

12 - Chapter A Eample : Find the center and radius of the circle 6 0. Graph the circle. Solution: To find the center and radius we must write the equation in standard form: Group and terms: 6 0 Move constant term to other side: 6 Complete the square: Rewrite in factored form: 9 Determine C and r: C, r Graph b plotting the center C, and appling the radius of units to find points on the circle: Since the radius is, the domain is, the range is,,. 0,6 and Eample 6: Find the center and radius of the circle 6 0. Solution: Although the coefficients of and are not, the equation represents a circle because the are equal, so we divide the equation b the common coeffient. 6 Divide equation b : 0 0 Group and terms: 0 Move constant term to other side: Complete the square: Rewrite in factored form: Determine C and r: C, r 9

13 - Chapter A Eercises for Chapter A - Rectangular Coordinate Sstem. Plot the following points on a rectangular coordinate sstem: a) A, b) B, c) C 0, d) D, e) E,0 f) F,. Use the points in eercise # to answer the following questions. a) Which point(s) are in quadrant I? quadrant II? quadrant III? quadrant IV? b) Which point(s) are on the -ais? the -ais? c) Which point(s) meet the condition: 0? d) Which point(s) meet the condition: 0? e) Which point(s) meet both the conditions: 0and?. Shade the region of the coordinate plane that contains each of the following sets of points. a), 0 b), and 0 c), and d), and. Graph the following relations and write their domain and range. a) 7,,,, 0,,, 6,,, 0,0, 9, b),,,,,.,.,,.6,.7. Write the domain and range of each of the relations in eercise. 6. Find the distance between the following sets of points. a) A, and B, b) C 0, and D, c) E,0 and F, d) G, e) J.,.9 and K., Is P 7, closer to point R, or point Q 9,? 8. Prove that the triangle with vertices A,, B,, andc 6,0 is a right triangle. 9. Find the area of triangle ABC in eercise #9. 0. Determine whether the triangle with vertices A,, B,, andc 9, is an isosceles triangle. (An isosceles triangle has two sides that are equal.). If the distance betweeen A, and B 6,0 is units, find all possible coordinates for A.. Write an equation that describes all the points, that are units from point B 6,0.. If the center of a circle is,9 and 0, is a point on the circle, find the radius of the circle.. Find the midpoint of the line segment connecting the following pairs of points: a) A, and B, b) C 0, and D, c) E,0 and F, d) G, and H, e) J.,.9 and K., 7.. In each of the following, M, is the mid-point of A and B. Find the missing end-point, A or B. a) A, ; M, b) M,0 ; B, 6. If 7, and,8 are endpoints of a diameter of a circle, find the center.

14 - Chapter A 7. Each of the following points A is an endpoint of a line segment. If the midpoint of the line segment AB is the point 0,0, findb. a), b), c), d), 6 e),0 8. Find the C h,k and radius r for the following circles. Then graph the circle. a) 7 6 b) 9 c) 7 d) 0 e) 8 0 f) Write an equation for each of the following circles. a) C, ; r b) C 0,0 ; r c) C, through 7, d) C 8, that touches the -ais at 0, 0. If,7 and, are endpoints of a diameter of a circle, write the equation for the circle.. Prove that the point C, is equidistant from A, and B 7,. Is C the midpoint of AC? Verif our answer.. If the diagonals (line segments connecting opposite vertices) of a parallelogram (a quadrilateral whose opposite sides are equal and parallel) are equal, the parallelogram is a rectangle (a parallelogram with four right angles). If all sides of the rectangle are equal, it is a square. Determine whether the quadrilateral with vertices A,, B,7, C,8, and D, is a parallelogram, rectangle, or square.

15 - Chapter A Answers to Eercises for Chapter A - Rectangular Coordinate Sstem. D(-,) A(-,-). a) QI: B; QII: D; QIII: A; QIV: F b) -ais: E; -ais: C c) B and F d) A, C, E, andf e) C and F. a) E(-,0) C(0,-) = 0 B(,) F(,-) b) = - = 0 c) = - = d)

16 - Chapter A = = = -. a) Domain: 7,,,,,9 Range:,,, 6,, b) Domain:,,,.,.6 Range:,,/,,.7. a) Domain: all real numbers 0 Interval notation:,0 Range: all real numbers Interval notation:, b) Domain: all real numbers greater than Interval notation:. Range: all real numbers 0 Interval notation: 0, c) Domain: all real numbers or, Range: all real numbers or, d) Domain: real numbers between and, including or, Range: real numbers or, 6. a) d A,B 6 6 Note: 6 is between 9 7 and 6 8. b) d C,D 0

17 - Chapter A. 669 c) d E,F d) d G,H. e) d J,K It s alwas helpful to graph the given situation on graph paper to help ou visualize the problem. Rememeber what formulas ou have to work with and what information each formula provides. To determine whether point R or point Q is closer to P, wemust determine the distance from R to P and from Q to P. The smaller of the two distances will tell us which point is closer. d P,R 6 d P,Q 0 Since d P,Q d P,R, Q is closer to P. 8. Plot the three points on graph paper, label them and draw the triangle. Remember that ou can prove that a triangle is a right triangle using the Pthagorean Theorem: a b c where a and b are the sides of the triangle and c is the hpotenuse. What formula we can use to find the length of each side? The distance formula, of course. d A,B 7 d B,C 68 7 d A,C 8 Once we have the lengths of each side, we plug the smaller sides into the Pthagoream Theorem for a and b, and the largest in for the hpotenuse Therefore, ΔABC is a right triangle. 9. To find the area of a triangle we can use the formula A bh, whereb is the base and h is the height. Note that h is the perpendicular distance from the third verte back to the base. Since ΔABC is a right triangle, the two legs are the base and the height, so that A TheareaofΔABC is 7square units. 0. Remember to plot the points and draw the triangle to help ou visualize the problem. To show that ΔABC is an isoscleles triangle we must show that two of the three sides are equal in length. Your plot above will probabl show ou which sides to tr first. d A,B 8 d B,C 8 Since sides AB and BC are equal in length ΔABC is an isosceles triangle.. Stating the problem algebraicall gives us d A,B. Replacing the left side with the distance formula, we get We must square both sides of the equation and solve for. 9 9

18 - Chapter A 9 6 Therefore A can be either of the points, and,. Justif this to ourself b plotting the points on graph paper.. d P,B Plot point B on graph paper. Sketch out the points that are units from point B. What shape do ou get? A circle.. Plot the center and point and sketch the circle. The radius of a circle r is the distance from the center of the circle to an point on the circle. Using the distance formula, we get r Find the midpoint of the line segment connecting the following pairs of points: a) Using the midpoint formula gives us M, M, b) M 0, M, c) M, M d) M,, M, e) M ,.,.6 M.0,.. a) Use Eample in the midpoint notes as a model for this problem. You should get the point 8, as our answer. b) 7, 6. Plot the given points on graph paper, draw the circle throught the two points, and draw the diameter. The diameter of a circle will alwas go through its center. Moreover, the center will be the mid-point of the diameter. To find the center, we must then find the midpoint of the two endpoints of the diameter using the midpoint formula. Center : 7, 8, 7. a), b), c), d),6 e),0 Note that the signs of the coordinates are opposites. If the origin is the midpoint of two points, we sa that the two points have smmetr with respect to the origin. 8. a) 7 6 Recall the formulas for circles. C,7 ; r b) C 0,0 ; r 7 c) C, ; r 7 d) C 0, ; r 0 e) Group -terms and move to other side of the equation: 8 Complete the square on the -terms b adding 8 6 to both sides of the equation Simplif: Therefore, C,0 ; r f) 0 0 0

19 - Chapter A C, ; r 0 9. a) b) c) We are given the center of the circle, so h andk, but we do not have the radius. Plot the center and draw the circle through P. What do we need to do to find the length of the radius? The distance formula! r d C,P 7 8 Substituting into the formula, we get 8 8 d) Plot the points on graph paper. You should be able to see that the radius of the circle is 8. Therefore the equation is The diameter of a circle goes through the center of a circle. Therefore, ou can use the midpoint formula to find the center of the circle. C, The radius is the distance from the center to either of the points on the circle. r. Plot out the problem on graph paper. Use the distance and midpoint formulas appropriatel to draw our conclusions algebraicall. You should find that C is equidistant from A and B, but it is not the midpoint. Eplain to ourself wh. Draw all the points that are equidistant from A and B. You should have drawn a line. This line is called the perpendicular bisector of line segment AB.. Using the distance formula appropriatel should show that all sides are equal in length and the diagonals are equal, therefore the most accurate term for the quadrilateral is square. Note that a square is also a parallelogram and a rectangle.

20 - Chapter B Chapter B - Graphs of Equations Graphing b Plotting Points We have seen that the coordinate sstem provides a method for locating points in a plane. Furthermore, we can plot sets of ordered pairs on the coordinate sstem to visualize the relationship between the two variables. However,in most cases we will be interested in relations that are stated as equations. For eample, the equation 6. refers to the relation, 6, read the set of all pairs, such that 6". Ever pair of numbers, that makes the equation true is called a solution to the equation. The pair, is a solution to the equation 6 because 6; but the ordered pair, is not a solution since 6. Question: Answer: How man solutions does the equation 6 have? There are an infinite number of solutions. Eample : Find three more pairs, such that 6. Solution: An pair of numbers whose sum is 6 is a solution. For eample:,,.,8.,,, 0,6, 6,0,,,, The plot of ever point that corresponds to a solution to the equation 6 is the graph of the equation 6: (-,) 0 (-.,8.) (.,.) (0,6) (,) (,) (6,0) The points labeled above are the solutions we listed in eample ; however, the graph consists of ever pair, that is a solution to the equation. Furthermore, ever point on the graph satisfies the equation 6.. Note that the above graph is a line. The graph of an linear equation is a line. A linear equation is one that can be written in the form a b c 0, where a and b are not both 0. The equation 6 can be written in the form of a linear equation 6 0, where a, b, and c 6. IMPORTANT CONCEPT: The graph of an equation consists of all pairs, that are solutions to the equation. Ever solution to the equation is a point on the graph and ever point on the graph, is a solution to the equation.

21 - Chapter B Eample : Which of the following pairs are points on the graph of 0? a),8 b), c), d) 0,0 e),9 Solution: a),8 is a solution because is a true statement. Therefore,8 is a point on the graph. b), is not a solution because Therefore,, is not on the graph. c), is not a solution because Therefore,, is not on the graph. d) 0,0 is a solution is a true statement. Therefore, 0,0 is a point on the graph. e),9 is a solution Therefore,,9 is a point on the graph. To find a solution to an equation, choose a value for, substitute it into the equation and solve for. Eample : Find three solutions to the equation 0. Solution: If, 0 0,0 is a solution. If, 0, is a solution. If, 0, is a solution. We can write this in chart form:, 0,0,, Note that each of the pairs, will be coordinates of points on the graph of 0.

22 - Chapter B Question: Is the point 0,0 on the graph of 0? Answer: 0,0 is not a point on the graph. For if it were, then the following equations would be true In fact, for the same reason, an point of the form 0, is not on the graph of the equation 0. Eample : Find fifteen solutions to the equation. Solution: Since we are going to substitute values for and solve for, it will be easier to first solve for : Substitute values for into, and solve for., 6 6 0,0,,,,,, 0 0 0,,,,,,,,,, 0,0 You should recognize that this equation gives us a graph that is a circle, and each of the points we found above is a point on the circle. Note that an values for that are less than or greater than will give us negative values under the radical sign. Therefore, -coordinates of the solutions will be in the interval,, which is the domain of the relation. To graph an equation b plotting points:. Solve the equation for.. Complete a table of values b substituting our choice of values for into the equation, then solving for. Use as man points as necessar to determine the shape of the graph.. Plot the points and draw a smooth curve through them..

23 - Chapter B Eample : Graph the equation. Solution: Solve for : Complete a table of values b picking an, and then using the above equation to determine the corresponding. Choose enough points to determine the shape of the graph. Plot the points and draw the curve., 6,6...,. 0,0 0...,. 0 0, 0...,. 0,0...,. 6, Note that the points on this graph do not form a straight line. This is because the equation is not in the form of a linear equation because is raised to the second power. Therefore, the graph will not be a line. This graph is called a parabola. The graph of a quadratic equation is a parabola. A quadratic equation is an equation that can be written in the form a b c, where a 0. The equation is a quadratic, where a, b 0, c. Note that an value for will give a real value for, so that the domain of the relation is,. The -values include all real numbers greater than or equal to, so that the range is,. Eample 6: Graph the equation. Solution: Solve for :. Complete a table of values. Plot the points and draw the curve.,, 0,0,,, 0 0, Note that the graph continues infinitel to the left and to the right, because we could place an number in the equation for and get a corresponding value for. Therefore, the domain of the relation is,. From the graph ou should be able to see that there are no -values larger than, so that the range is,

24 - Chapter B Eample 7: Graph the equation. Solution: Solve for :. Complete a table of values. Plot the points and draw the curve., 0 no solution no solution 0,0. 0.., 0.,, Note that the graph continues infinitel to the right because we could place an number in the equation for that is greater than and get a corresponding value for. Question: What happens to the graph when? Answer: When numbers were substituted for that were less than, we could not compute because we had a negative value under the radical. Therefore the graph does not etend to the left of the point,0. Since must be greater than or equal to for to be a real number, the domain of the relation is,. B eamining the graph and table of values, note that the -coordinates are greater than or equal to 0, so that the range is 0,

25 - Chapter B Intercepts Points that will prove to be ver important to our future stud of equations and their graphs are the intercepts. Definition: The points where the graph of an equation crosses the -ais is called the -intercept. If a,0 is a point on the graph of an equation, we sa that a is an -intercept of the graph because a,0 is a point on the -ais. Note that the -intercept will alwas have a -coordinate of 0. The -intercepts are also called zeros of the equation. A zero is a value for that causes the corresponding -value to be 0. If the point,0 is an -intercept for the graph of an equation, isazero of the equation. Definition: The points where the graph of an equation crosses the -ais is called the -intercept. If 0,b is a point on the graph of an equation, we sa that b is a -intercept of the graph because 0,b is a point on the -ais. Note that the -intercept will alwas have an -coordinate of 0. B inspecting the table of values and the graph of, we see the equation crosses the -ais at the point 0, and the -ais at the points,0 and,0. Therefore, isthe-intercept and and are-intercepts (,0) (0,-) Finding the -intercepts To find the -intercept(s) of an equation, substitute 0 for, and solve for. Finding the -intercepts To find the -intercept(s) of an equation, substitute 0 for, and solve for. Eample : Find the -and-intercepts of algebraicall. Solution: Find the intercepts b substituting 0 for the appropriate variable. -intercept: Substitute 0 for and solve for : 0 is the -intercept, and 0, is a point on the graph. -intercept: Substitute 0 for and solve for : and andare-intercepts and,0 and,0 are points on the graph.

26 - Chapter B When graphing b plotting points, it is important to include points close to andoneithersideofthe-intercepts. Question: Wh is the above true? Answer: The -intercepts are the zeros that is, the are the values for that cause 0. The -values ma change sign on either side of a zero which means that a graph ma be below the -ais on one side of an -intercept and above it on the other, or vice-versa. Numbers close to zero often behave differentl. For eample, if 0,. Prove this to ourself using several fractions that are between 0 and. E: Eample : Find the intercepts of the graph of. Solution: -intercepts: Substitute 0 for and solve for : 0 Thus, andare-intercepts; 0, and 0, are points on the graph. -intercepts: Substitute 0 for and solve for : 0 Thus, andare-intercepts;,0 and,0 are points on the graph. We can confirm our findings b inspecting the graph of which we know to be a circle with C 0,0 and radius. (-,0) 6 (0,) (0,-) (,0) Eample : Find the intercepts of 6. Solution: -intercepts: Substitute 0 for and solve for : 0, Thus, the point 0, 6 is the -intercept. -intercepts: Substitute 0 for and solve for :, and 0 6 and Thus, the two points 6,0 and,0 are the -intercepts.

27 - Chapter B Eample : Find the intercepts of 6 8. Solution: -intercepts: Substitute 0 for and solve for : 0, / Thus, the -intercept is the point 0,. -intercepts: Substitute 0 for and solve for :, and 0 and Thus, the -intercepts are the points,0 and.0.

28 - Chapter B Smmetr When we think of smmetr in art, we think of balance on both sides of a center of focus. In other situations, the term conjures up the idea of a mirror image. In either case, the notion of smmetr means essentiall the same thing in an algebra class: a shape looks the same on both sides of a dividing line or point. The dividing line is called the line of smmetr. We will also consider points of smmetr in this section. Eamine the graph of the circle for smmetr Question: How man lines or points of smmetr can ou find for the circle? Answer: The circle has smmetr about its center a point, and about an straight line that goes through its center an infinite number of lines. Although an line that goes through the center of the circle can be a line of smmetr for the circle, we will focus our efforts in this section on the -and-aes. We will also look at smmetr about the origin. Eample : A, B, andc. The figures on the aes below are smmetric about the -ais. Find the coordinates of Solution: A, B, 7 C, Note that the -coordinates are the same as the corresponding point in the Quadrant I figure, but the -coordinates are the negatives of the -coordinates in the Quadrant I figure (, ) A (, ) C (, 7) B Eample : Find the points of the figure that would be smmetric, about the -ais, to the above figure in Quadrant I. Solution: Corresponding points would be,,7,. Note that the -coordinates of corresponding points are the same as the Quadrant I figure, but the -coordinates are the negatives.

29 - Chapter B Smmetric about the -ais. When we sa that a graph is smmetric with respect to the -ais we mean that if the graph is reflected through or reflected about the -aiswegetthesamegraph. Note that the graph of the equation is smmetric about the -ais. That is, if we reflected the graph 80 about the -ais, the graph would be the same. See the plot below. Eamine the table of values that we used to graph = Question: What do ou notice about the -values when and? What about the values for. and.? Answer: For an number and its opposite, the -values are equal. Definition: A graph is smmetric about the -ais if and onl if for ever point, on the graph,, will also be a point on the graph. This definition leads us to the following test for smmetr about the -ais. Test for Smmetr about the -ais. To determine algebraicall if a graph will be smmetric about the -ais, substitute for and simplif. If the resulting equation is equivalent to the original equation, the graph will be smmetric about the -ais. Eample : Show that has smmetr with respect to the -ais. Solution: Substituting for in the original equation, we get. Simplifing,. The result is equivalent to the original equation; therefore, the graph of is smmetric about the -ais. Eample : Test the equation for smmetr about the -ais. Solution: Substitute for amd simplif Since the resulting equation is the same as the original, the graph is smmetric about the -ais.

30 - Chapter B The net tpe of smmetr we discuss is that of a graph being smmetric with respect to the -ais. The following graph has smmetr about the -ais. That is, the graph will be the same if it is reflected 80 about the -ais. For this to occur, ever must be paired with both and. (, ) (, ) (, - ) (, - ) The graph will be the same if it is reflected 80 about the -ais. Definition: A graph has smmetr about the -ais if and onl if, is a point on the graph for ever point, on the graph. Test for Smmetr about the -ais: Substitute for in the equation and simplif. If the resulting equation is equivalent to the original, the graph will be smmetric about the -ais. Eample : Solution: Test for smmetr about the -ais. Substitute for and simplif Multipl both sides b : Since neither of the resulting equations is equivalent to the original, the graph of the equation is not smmetric about the -ais. The following graph has smmetr about the origin. That is, it will be the same if it is rotated 80 about the origin. Note that for ever point,, the point, will also be on the graph. Furthermore, the line segment connecting these corresponding pairs will go through the origin. (-, ) (, ) (-, -) - - (, -) - - Definition: The graph of an equation will have smmetr about the origin if and onl if for ever point, on the graph, the point, will also be on the graph.

31 - Chapter B Test for Smmetr about the origin: Substitute for, and for into the equation. If the resulting equation is equivalent to the original, the graph has smmetr about the origin. Eample 6: Test for smmetr about the origin. Solution: Substitute for, and for, and simplif. Since the resulting equation is not equivalent to the original equation, there is no smmetr about the origin. Note that even if we multipl the resulting equation b, the result is not equivalent to the original equation. The results of the eamples in this section have shown that has smmetr about the -ais onl. Eample 7: Test the equation for smmetr about the -ais, -ais, and origin. Solution: -ais: [Substitute for ]: Since the resulting equation is equivalent to the original, the graph has smmetr about the -ais. -ais: [Substitute for ]:. equation has smmetr about the -ais. origin: [Substitute for and for ]: equation has smmetr about the origin. -

32 - Chapter B Eample 8: Test the equation for smmetr about the -ais, -ais, and origin. Solution: -ais: [Substitute for ]: equation does not have smmetr about the -ais. -ais: [Substitute for ]: equation does not have smmetr about the -ais. origin: [Substitute for and for ]: equation has smmetr about the origin Knowing whether the graph of a particular equation has smmetr about a point or line can assist us if we are graphing b plotting points. If we are using a grapher it provides us information about what our graph should look like so that we can determine whether a particular graph is reasonable for the equation we entered. Consider the section of graph in the figure below. If we knew that the graph had smmetr about the -ais, -ais, or origin, we could complete the graph knowing onl what the Quadrant I section of the graph looked like. Eample 9: Reproduce the graph above using paper and pencil. Complete the graph for each of the following situations. The graph has smmetr about the a) -ais b) -ais c)origin. Solution: (a) (b) (c)

33 - Chapter B To use smmetr in graphing an equation,. Determine whether the graph has smmetr about the -ais, -ais, or origin.. Plot enough points in the first quadrant to get an idea of what the graph looks like.. Appl smmetr to complete the graph. Eample 0: Use smmetr in graphing the equation. Solution: Determine smmetr. -ais: No smmetr about -ais. -ais: No smmetr about -ais. origin: Smmetr about the origin. Because has smmetr about the origin, use values of that are positive in the table of values, and use smmetr to find the remainder of the graph. / / Since, for eample, the point /, is on the graph of, and this graph is smmetric about the origin, we know that the point /, is also a point on the graph of. -

34 - Chapter B Eercises for Chapter B - Graphs of Equations. Graph. Graph 6.. Graph.. Graph.. Graph. 6. Graph. 7. Graph. 8. Find the intercepts for Find the intercepts for Find the intercepts for.. Find the intercepts for.. Find the intercepts for.. Find the intercepts for.. Test each equation for smmetr about the -ais, -ais, or origin. a) 0 b) 0 c) 9. d) 6 e) f). Use smmetr about the -ais, -ais, or origin to sketch the graph of each. a) b) c) d) 6. In one of the previous eamples we plotted some points on the graph of,and did not plot an points for which was positive. Find values of which satisf this equation for /0. Hint this equation is a quadratic in.

35 - Chapter B Answers to Eercises for Chapter B - Graphs of Equations (, ) (, -)

36 - Chapter B 6. (-, ) (, -) 7. (-, -) (, ) 8. -intercept: Substitute 0: or 0, -intercept: Substitute 0: or,0 9. -intercept: Substitute 0: or 0,6 -intercept: Substitute 0: 0 6 0, or,0,,0 0. -intercept: Substitute 0: 0 or 0, -intercept: Substitute 0: 0 or,0,,0. -intercept: Substitute 0: 0 or 0, -intercept: Substitute 0: no -intercepts. -intercept: Substitute 0: or 0, -intercept: Substitute 0: , or,0,,0. -intercept: Substitute 0: 0 no -intercepts -intercept: Substitute 0: 0 0 no -intercepts. a)-ais: Substitute for : 0 0 No smmetr -ais: Substitute for : 0 0 No smmetr Origin: Substitute for and for : 0 0 Multipling both sides b : 0 Smmetr Conclusion: Smmetr about the origin onl. b) -ais: Substitute for : 0 0 No smmetr -ais: Substitute for : 0 0 Smmetr Origin: Substitute for and for : 0 0 No smmetr Conclusion: Smmetr about the -ais onl. c) -ais: Substitute for : 9 9 No smmetr -ais: Substitute for : 9 9 No smmetr Origin: Substitute for and for : 9 9

37 - Chapter B Multipling both sides b : 9 Smmetr Conclusion: Smmetr about the origin onl. d) -ais: Substitute for : 6 6 Smmetr -ais: Substitute for : 6 6 No smmetr Origin: Substitute for and for : 6 6 No smmetr Conclusion: Smmetr about the -ais onl. e) -ais: Substitute for : No smmetr -ais: Substitute for : Smmetr Origin: Substitute for and for : No smmetr Conclusion: Smmetr about the -ais onl. f) Smmetr about the -ais, -ais, and origin. You prove it!. a) smmetr about the -ais onl (0, -) b) smmetr about the -ais,, c) Smmetr about the -ais, -ais, and origin. - - d) Smmetr about the origin. (-, ) (, -)

38 - Chapter B 6. If, then we have Setting 0 we have /0 0 Thus, /

39 - Chapter C Chapter C - Linear Equations in Two Variables Linear Equations in Two Variables In this section we will eamine equations in and where both variables are raised to the first power. Definition: An equation that can be written in the form A B C, wherea and B are not both 0, is called a linear equation. Question: According to the definition, which of the following are linear equations? a) b) 0 c) d) e) Answer: a) Yes. A, B, C. and are raised to the first power. b) Yes. 0 can be written as 0, so that A, B 0, C. A and B are not both 0, the variables are raised to the first power. c) Yes. can be written as. A, B, C. Both variables are raised to the first power. d) No. is raised to the second power. e) No. is raised to the second power. Linear equations can be graphed as an other equation b plotting points. Eample : Graph the linear equation. Solution: Solve for. Fill in a table of values, 7,7, 0 0,,, Plot the points and sketch (-,7) (-,) (0,) (,) (,-) Notice that all of the points were on a line. You might guess that this is wh these equations are called linear equations. The graph of an linear equation is a straight line. Because an two points determine a line, we can graph a linear equation b plotting at least two points. The intercepts are a good choice because the are usuall eas points to find. Question: What are the -and-intercepts of the above graph? Answer: -intercept: Substitute 0. 0 or 0, -intercept: Substitute 0. 0 or,0

40 - Chapter C Eample : Graph the linear equation 8. Solution: Solve for : 8 Make a table of values: Plot the points and draw the line:,, 0 0, 7,7 0,0 (-,) (?,0) - (,0) (,6) (0,) We can see from both the table and the graph that the -intercept is. Question: What is the -intercept? Answer: If 0, The -intercept is 8. Eample : Graph the linear equation. Solution: We recognize that is a linear equation. Make a table of values: Plot the points and draw the line:, 0 0, 0,0, (0,) (,) (-/, 0) Eample : Graph the linear equation 6 0. Solution: We recognize that 6 0 is a linear equation. Solving for :. Note that the equation does not contain a variable. The onl condition that eists for a point to be included as a solution to the equation is that must be. Make a table of values: Plot the points and draw the line:, 0 0,,, All points with -coordinate lie on the horizontal line units below the -ais.

41 - Chapter C In the following investigation ou ma use a graphing calculator but record the table of values and the graph of each equation on graph paper before graphing the net equation. As ou graph each equation, observe the results before graphing the net equation. If ou have a TI-8 or TI-8, ou can enter the equation using the Y buttononthetopleft of the displa. (You must solve the equation for first.) Set the viewing window using the WINDOW to the right of the Y button. The following viewing window gives a FRIENDLY WINDOW: min.7, ma.7. scl tells ou how man units ou want each tick mark to represent. min., ma. will give a SQUARE WINDOW. Otherwise, the range values can be manipulated to include the part of the graph ou want to see. You can multipl each of the above settings to get other friendl and square windows. If ou do not use a friendl window, ou will get large decimals when ou trace the graph using the TRACE button. Use right and left arrows to move the cursor over the graph. The and at the bottom of the screen will give the coordinates of the point highlighted b the cursor. The cursor ma skip over the integral -intercepts if the window is not a friendl window. Eample : Graph the following lines on the same coordinate sstem. a) Which number was different in each equation? What effect did this have on the graph? b) Which number was the same in each equation? What did each line have in common? Solution: a) The coefficient of was different in each equation. The coefficient of affected the steepness of the graph. b) The constant was in each equation. The all had a -intercept of. Eample 6: Graph the following lines on the same coordinate sstem. a) Which number was different in each equation? What effect did this have on the graph? b) What number was the same in each equation? What did each line have in common? Solution: a) The constant was different in each equation. The -intercepts were different in each line. b) The coefficient of was in each equation. Each line had the same steepness. Eample 7: Graph the following lines on the same coordinate sstem and compare with the lines in Eample. What is different between the equation of the lines in this eample and the lines in Eample? What effect did this difference have on the graphs? Solution: The coefficients of in Eample are positive, but the are negative in this eample. Graphs in Eample go uphill from left to right. Graphs in this eample go downhill from left to right. Conclusions: (Graphs of lines in the form m b. m, the coefficient of, affects the steepness of the line. The larger m, the steeper the line. When m 0, the line goes uphill from left to right. We sa the line is increasing. When m 0, the line goes downhill from left to right. We sa the line is decreasing. The number b is the -intercept. If 0, then m 0 b b

42 - Chapter C Slopes of Lines The idea of steepness has applications in man areas of our lives. When we build a house, we must determine the steepness of a roof called the pitch of the roof. Certainl if we are skiers or cclists, we are interested in the steepness of a ski slope or roadwa. In some cases it is important that we be able to measure steepness. For eample, the Americans with Disabilities Act requires that a wheel chair ramp rise no more than 6 inches for 6 feet of horizontal distance, as shown below. Ramp 6in.=.ft. 6 ft. In the above eample, we would call the vertical change the rise and the horizontal distance the run. We describe the steepness as the ratio of the rise to the run. Consider the line below that goes through the points A, and B 6, (,) (6,) 6 - = run (6,) } rise - = The rise can be calculated b subtracting the -coordinates of the given points and the run can be calculated b subtracting the -coordinates of the points. The ratio of the rise to the run can be simplified to. This ratio is called the slope of the line. Definition: The slope of a line, m, is the ratio of the change in to the change in. m rise change in Δ run change in Δ The following formula generalizes the process we used above to calculate the slope of an line if we are given two points on the line: P, and P,. The rise Δ and the run Δ can be calculated as follows: Δ and Δ. Definition: The slope of the line through points P, and P, is m. Eample : Plot each pair of points on graph paper and find the slope of the line: a), and, b),7 and,7 c), and,6 d), and, Solution: a) m 7 7 b) m c) m d) m 6 0 undefined

43 - Chapter C Eample : Use our results to complete each sentence. a) The line goes uphill from left to right when the slope is. b) The line goes downhill from left to right when the slope is. c) The line is horizontal when the slope is. d) The line is vertical when the slope is. Answers: a) positive b) negative c) 0 d) undefined Investigation of Slopes with a Grapher Enter the equation into our grapher. Use TABLE mode to answer the following questions. Question: How much does change ever time increases b unit? Answer: units. For eample when changes from 0 to, changes from to. Quesiton: How much does change ever time increases b units? Answer: changes 6 units. For eample, when changes from 0 to, changes from to 7, a total of 6 units. Question: How much does change ever time increases b units? Answer: 9 units. For eample, when changes from to, changes from 7 to 6, a total of 9 units. change in Question: Write each of the above answers as a ratio: change in. Are these ratios equivalent? Answer: 6 9. All these ratios are equivalent to. Quesiton: Compare the slope of the line and the coefficient of in the equation of the line. Answer: The slope of the line is. The coefficient of in the equation of the line is. Eample : Enter the equation into our grapher. a) Use two points from our table to calculate the slope of the line. Compare with the coefficient of in the equation. b) Tr several more eamples of lines in the form m b. What can ou conclude about the relationship between the coefficient of of a linear equation that has been solved for and the slope of the line? Solution: a) m which is also the coefficient of in the equation. b) The coefficient of is the slope of the line. Conclusions: The coefficient of is the slope of the line. If the slope is positive, the graph goes uphill from left to right. We sa the line is increasing. If the slope is negative, the graph goes downhill from left to right. We sa the line is decreasing.

44 - Chapter C Not onl can we find the slope of a line b solving its equation for, but the form m b is especiall hand in graphing lines. We must use the " " form to enter the equation into the grapher, but it also makes pencil and paper graphing easier, as well. To graph a line using the m b form, Since the -intercept is b, plot the point 0,b on the -ais. change in From the -intercept, appl the slope, m change in rise run to locate a second point on the line. Draw the line. Eample : Graph. Solution: The -intercept is, so we plot the point 0,. The slope is which means that decreases units when increases unit. We then count one unit to the right and units down from the -intercept to plot our second point, and draw the line m = run= rise = Since the slope of the line is negative, we epect the line to go downhill from left to right. We sa this line is decreasing. Eample : Find the slope and -intercept of the line 6. Then graph. Solution: We must first solve the equation for : 6 6. Therefore, the slope m and the -intercept b. We first plot the point 0,. From that point we use the slope to rise and run. Plot the second point and draw the line rise run (0, -) We epect the line to go uphill from left to right since the slope is positive. We sa this line is increasing.

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

Coordinate Geometry. Positive gradients: Negative gradients:

Coordinate Geometry. Positive gradients: Negative gradients: 8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT

SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT . Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

3 Rectangular Coordinate System and Graphs

3 Rectangular Coordinate System and Graphs 060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it

10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it 3330_00.qd /8/05 9:00 AM Page 735 Section 0. Introduction to Conics: Parabolas 735 0. Introduction to Conics: Parabolas What ou should learn Recognize a conic as the intersection of a plane and a double-napped

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

Chapter 7: Eigenvalues and Eigenvectors 16

Chapter 7: Eigenvalues and Eigenvectors 16 Chapter 7: Eigenvalues and Eigenvectors 6 SECION G Sketching Conics B the end of this section ou will be able to recognise equations of different tpes of conics complete the square and use this to sketch

More information

SLOPES AND EQUATIONS OF LINES CHAPTER

SLOPES AND EQUATIONS OF LINES CHAPTER CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8- The Slope of a Line 8- The Equation of a Line 8-3 Midpoint of a Line Segment 8-4 The Slopes of Perpendicular Lines 8-5 Coordinate Proof 8-6 Concurrence of the

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

The Rectangular Coordinate System

The Rectangular Coordinate System The Mathematics Competenc Test The Rectangular Coordinate Sstem When we write down a formula for some quantit,, in terms of another quantit,, we are epressing a relationship between the two quantities.

More information

SECTION 9-1 Conic Sections; Parabola

SECTION 9-1 Conic Sections; Parabola 66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships

More information

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. Plot the following points on the real. P2. Determine which of the following are solutions Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

More information

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1 Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

More information

Lesson 8.3 Exercises, pages

Lesson 8.3 Exercises, pages Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs 6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Quadratic Functions. MathsStart. Topic 3

Quadratic Functions. MathsStart. Topic 3 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

More information

Transformations of Function Graphs

Transformations of Function Graphs - - - 0 - - - - - - - Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

Are You Ready? Circumference and Area of Circles

Are You Ready? Circumference and Area of Circles SKILL 39 Are You Read? Circumference and Area of Circles Teaching Skill 39 Objective Find the circumference and area of circles. Remind students that perimeter is the distance around a figure and that

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

Linear Equations in Two Variables

Linear Equations in Two Variables Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

More information

Translating Points. Subtract 2 from the y-coordinates

Translating Points. Subtract 2 from the y-coordinates CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

More information

1.6 Graphs of Functions

1.6 Graphs of Functions .6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

THE PARABOLA 13.2. section

THE PARABOLA 13.2. section 698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

More information

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form. Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

More information

3 Functions and Graphs

3 Functions and Graphs 54617_CH03_155-224.QXP 9/14/10 1:04 PM Page 155 3 Functions and Graphs In This Chapter 3.1 Functions and Graphs 3.2 Smmetr and Transformations 3.3 Linear and Quadratic Functions 3.4 Piecewise-Defined Functions

More information

x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m = Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the

More information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane.

INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane. P ( 1, 1 ) INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing Distance Between Two Points On The Coordinate Plane Q (, ) R (, 1 ) The straight line distance between two points P ( 1, 1

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

Linear Inequality in Two Variables

Linear Inequality in Two Variables 90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.

More information

THE PARABOLA section. Developing the Equation

THE PARABOLA section. Developing the Equation 80 (-0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or

More information

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1) Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

More information

Rising Algebra 2 Honors

Rising Algebra 2 Honors Rising Algebra Honors Complete the following packet and return it on the first da of school. You will be tested on this material the first week of class. 50% of the test grade will be completion of this

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

Perimeter and Area of Geometric Figures on the Coordinate Plane

Perimeter and Area of Geometric Figures on the Coordinate Plane Perimeter and Area of Geometric Figures on the Coordinate Plane There are more than 200 national flags in the world. One of the largest is the flag of Brazil flown in Three Powers Plaza in Brasilia. This

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it 0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

Rational Functions, Equations, and Inequalities

Rational Functions, Equations, and Inequalities Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

TRANSFORMATIONS AND THE COORDINATE PLANE

TRANSFORMATIONS AND THE COORDINATE PLANE CHPTER TRNSFORMTIONS ND THE COORDINTE PLNE In our stud of mathematics, we are accustomed to representing an equation as a line or a curve. This blending of geometr and algebra was not alwas familiar to

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math

2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math Chapter 2 Lesson 2-5 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular

More information

5.3 Graphing Cubic Functions

5.3 Graphing Cubic Functions Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

More information

Analyzing the Graph of a Function

Analyzing the Graph of a Function SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

More information

If (a)(b) 5 0, then a 5 0 or b 5 0.

If (a)(b) 5 0, then a 5 0 or b 5 0. chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

More information

Modifying Functions - Families of Graphs

Modifying Functions - Families of Graphs Worksheet 47 Modifing Functions - Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and

More information

1.4 RECTANGULAR COORDINATES, TECHNOLOGY, AND GRAPHS

1.4 RECTANGULAR COORDINATES, TECHNOLOGY, AND GRAPHS .4 Rectangular Coordinates, Technolog, and Graphs 5 34. What is the smallest integer that is greater than 348 37 35. What is the smallest even integer that is greater than 5? 36. What is the largest prime

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

Section 2.1 Rectangular Coordinate Systems

Section 2.1 Rectangular Coordinate Systems P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

Trigonometry Review Workshop 1

Trigonometry Review Workshop 1 Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

More information

A Summary of Curve Sketching. Analyzing the Graph of a Function

A Summary of Curve Sketching. Analyzing the Graph of a Function 0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph

More information

NCERT. The coordinates of the mid-point of the line segment joining the points P (x 1. which is non zero unless the points A, B and C are collinear.

NCERT. The coordinates of the mid-point of the line segment joining the points P (x 1. which is non zero unless the points A, B and C are collinear. COORDINATE GEOMETRY CHAPTER 7 (A) Main Concepts and Results Distance Formula, Section Formula, Area of a Triangle. The distance between two points P (x 1 ) and Q (x, y ) is ( x x ) + ( y y ) 1 1 The distance

More information

Chapter 12. The Straight Line

Chapter 12. The Straight Line 302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

More information