DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS


 Derick Rodgers
 7 years ago
 Views:
Transcription
1 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 plane. If, as in Figure G., we form a right triangle with P and P as vertices, then it follows from Theorem B.4 in Appendi B that the sides of that triangle have lengths and. Thus, it follows from the Theorem of Pthagoras that d = + = ( ) + ( ) and hence we have the following result. G. theorem. The distance d between two points P (, ) and P (, ) in a coordinate plane is given b d = ( ) + ( ) () P (, ) d P (, ) Figure G. To appl Formula () the scales on the coordinate aes must be the same; otherwise, we would not have been able to use the Theorem of Pthagoras in the derivation. Moreover, when using Formula () it doesnot matter which point islabeled P and which one islabeled P, since reversing the pointschangesthe signsof and ; thishasno effect on the value of d because these quantities are squared in the formula. When it is important to emphasize the points, the distance between P and P isdenoted b d(p,p ) or d(p,p ). Eample Find the distance between the points (, 3) and (, 7). Solution. If we let (, ) be (, 3) and let (, ) be (, 7), then () ields A6 d = [ ( )] +[7 3] = = 5 = 5
2 Appendi G: Distance, Circles, and Quadratic Equations A63 A(4, 6) B(, 3) Figure G. C(7, 5) Eample It can be shown that the converse of the Theorem of Pthagoras is true; that is, if the sides of a triangle satisf the relationship a + b = c, then the triangle must be a right triangle. Use this result to show that the points A(4, 6), B(, 3), and C(7, 5) are vertices of a right triangle. Solution. The points and the triangle are shown in Figure G.. From (), the lengths of the sides of the triangles are Since d(a,b) = ( 4) + ( 3 6) = = 90 d(a,c) = (7 4) + (5 6) = 9 + = 0 d(b,c) = (7 ) +[5 ( 3)] = = 00 = 0 [d(a,b)] +[d(a,c)] =[d(b,c)] it follows that ABC is a right triangle with hpotenuse BC. a Figure G.3 Figure G.4 (b a) P (, ) b a M(, ) b P (, ) THE MIDPOINT FORMULA It is often necessar to find the coordinates of the midpoint of a line segment joining two points in the plane. To derive the midpoint formula, we will start with two points on a coordinate line. If we assume that the points have coordinates a and b and that a b, then, as shown in Figure G.3, the distance between a and b is b a, and the coordinate of the midpoint between a and b is a + (b a) = a + b = (a + b) which is the arithmetic average of a and b. Had the points been labeled with b a, the same formula would have resulted (verif). Therefore, the midpoint of two points on a coordinate line is the arithmetic average of their coordinates, regardless of their relative positions. If we now let P (, ) and P (, ) be an two points in the plane and M(,) the midpoint of the line segment joining them (Figure G.4), then it can be shown using similar triangles that is the midpoint of and on the ais and is the midpoint of and on the ais, so = ( + ) and = ( + ) Thus, we have the following result. G. theorem (The Midpoint Formula). The midpoint of the line segment joining two points (, ) and (, ) in a coordinate plane is ( ( + ), ( + ) ) () r ( 0, 0 ) (, ) Eample 3 Find the midpoint of the line segment joining (3, 4) and (7, ). Solution. From () the midpoint is ( (3 + 7), ( 4 + )) = (5, ) Figure G.5 CIRCLES If ( 0, 0 ) is a fied point in the plane, then the circle of radius r centered at ( 0, 0 ) is the set of all points in the plane whose distance from ( 0, 0 ) is r (Figure G.5). Thus, a point
3 A64 Appendi G: Distance, Circles, and Quadratic Equations (, ) will lie on this circle if and onl if ( 0 ) + ( 0 ) = r or equivalentl, This is called the standard form of the equation of a circle. ( 0 ) + ( 0 ) = r (3) Eample 4 Find an equation for the circle of radius 4 centered at ( 5, 3). Solution. From (3) with 0 = 5, 0 = 3, and r = 4 we obtain ( + 5) + ( 3) = 6 If desired, this equation can be written in an epanded form b squaring the terms and then simplifing: ( ) + ( 6 + 9) 6 = = 0 Eample 5 (4, ). Find an equation for the circle with center (, ) that passes through Solution. The radius r of the circle is the distance between (4, ) and (, ),so r = ( 4) + ( ) = 5 We now know the center and radius, so we can use (3) to obtain the equation ( ) + ( + ) = 5 or = 0 FINDING THE CENTER AND RADIUS OF A CIRCLE When ou encounter an equation of form (3), ou will know immediatel that its graph is a circle; its center and radius can then be found from the constants that appear in the equation: ( 0 ) + ( 0 ) = r coordinate of the center is 0 coordinate of the center is 0 radius squared + = Eample 6 equation of a circle center ( 0, 0 ) radius r ( ) + ( 5) = 9 ( + 7) + ( + ) = 6 + = 5 ( 4) + = 5 (, 5) ( 7, ) (0, 0) (4, 0) Figure G.6 The unit circle The circle + =, which is centered at the origin and has radius, is of special importance; it is called the unit circle (Figure G.6). OTHER FORMS FOR THE EQUATION OF A CIRCLE An alternative version of Equation (3) can be obtained b squaring the terms and simplifing. This ields an equation of the form + + d + e + f = 0 (4) where d, e, and f are constants. (See the final equations in Eamples 4 and 5.)
4 Appendi G: Distance, Circles, and Quadratic Equations A65 Still another version of the equation of a circle can be obtained b multipling both sides of (4) b a nonzero constant A. This ields an equation of the form A + A + D + E + F = 0 (5) where A, D, E, and F are constants and A = 0. If the equation of a circle is given b (4) or (5), then the center and radius can be found b first rewriting the equation in standard form, then reading off the center and radius from that equation. The following eample shows how to do this using the technique of completing the square. In preparation for the eample, recall that completing the square is a method for rewriting an epression of the form + b as a difference of two squares. The procedure is to take half the coefficient of, square it, and then add and subtract that result from the original epression to obtain + b = + b + (b/) (b/) =[ + (b/)] (b/) Eample 7 Find the center and radius of the circle with equation (a) = 0 (b) = 0 Solution (a). right side: First, group the terms, group the terms, and take the constant to the ( 8) + ( + ) = 8 Net we want to add the appropriate constant within each set of parentheses to complete the square, and subtract the same constant outside the parentheses to maintain equalit. The appropriate constant is obtained b taking half the coefficient of the firstdegree term and squaring it. This ields ( 8 + 6) 6 + ( + + ) = 8 from which we obtain ( 4) + ( + ) = or ( 4) + ( + ) = 9 Thus from (3), the circle has center (4, ) and radius 3. Solution (b). The given equation is of form (5) with A =. We will first divide through b (the coefficient of the squared terms) to reduce the equation to form (4). Then we will proceed as in part (a) of this eample. The computations are as follows: = 0 We divided through b. ( + ) + = 8 ( ) + = 8 ( + 6) + = From (3), the circle has center ( 6, 0) and radius We completed the square. DEGENERATE CASES OF A CIRCLE There is no guarantee that an equation of form (5) represents a circle. For eample, suppose that we divide both sides of (5) b A, then complete the squares to obtain ( 0 ) + ( 0 ) = k Depending on the value of k, the following situations occur:
5 A66 Appendi G: Distance, Circles, and Quadratic Equations (k > 0) The graph is a circle with center ( 0, 0 ) and radius k. (k = 0) The onl solution of the equation is = 0, = 0, so the graph is the single point ( 0, 0 ). (k < 0) The equation has no real solutions and consequentl no graph. Eample 8 Describe the graphs of (a) ( ) + ( + 4) = 9 (b) ( ) + ( + 4) = 0 Solution (a). There are no real values of and that will make the left side of the equation negative. Thus, the solution set of the equation is empt, and the equation has no graph. Solution (b). The onl values of and that will make the left side of the equation 0 are =, = 4. Thus, the graph of the equation is the single point (, 4). The following theorem summarizes our observations. The last two cases in Theorem G.3 are called degenerate cases. In spite of the fact that these degenerate cases can occur, (6) isoften called the general equation of a circle. G.3 theorem. An equation of the form A + A + D + E + F = 0 (6) where A = 0, represents a circle, or a point, or else has no graph. THE GRAPH of = a + b + c An equation of the form = a + b + c (a = 0) (7) is called a quadratic equation in. Depending on whether a is positive or negative, the graph, which is called a parabola, has one of the two forms shown in Figure G.7. In both cases the parabola is smmetric about a vertical line parallel to the ais. This line of smmetr cuts the parabola at a point called the verte. The verte is the low point on the curve if a>0 and the high point if a<0. Verte b/(a) Verte b/(a) Figure G.7 = a + b + c a > 0 = a + b + c a < 0 In the eercises (Eercise 78) we will help the reader show that the coordinate of the verte is given b the formula = b (8) a
6 Appendi G: Distance, Circles, and Quadratic Equations A = 3 With the aid of this formula, a reasonabl accurate graph of a quadratic equation in can be obtained b plotting the verte and two points on each side of it. Eample 9 Sketch the graph of (a) = (b) = Solution (a). The equation is of form (7) with a =, b =, and c =, so b (8) the coordinate of the verte is = b a = Using this value and two additional values on each side, we obtain Figure G.8. Figure G Figure G.9 = = = Solution (b). The equation is of form (7) with a =, b = 4, and c = 5, so b (8) the coordinate of the verte is = b a = Using this value and two additional values on each side, we obtain the table and graph in Figure G.9. Quite often the intercepts of a parabola = a + b + c are important to know. The intercept, = c, results immediatel b setting = 0. However, in order to obtain the intercepts, if an, we must set = 0 and then solve the resulting quadratic equation a + b + c = 0. Eample 0 Solve the inequalit > 0 Solution. Because the left side of the inequalit does not have readil discernible factors, the testpoint method illustrated in Eample 4 of Appendi A is not convenient to use. Instead, we will give a graphical solution. The given inequalit is satisfied for those values of where the graph of = is above the ais. From Figure G.8 those are the values of to the left of the smaller intercept or to the right of the larger intercept. To find these intercepts we set = 0 to obtain = 0 Solving b the quadratic formula gives = b ± b 4ac = ± = ± 3 a Thus, the intercepts are = and = and the solution set of the inequalit is (, 3) ( + 3, + ) Note that the decimal approimationsof the interceptscalculated in the preceding eample agree with the graph in Figure G.8. Observe, however, that we used the eact values of the intercepts to epressthe solution. The choice of eact versusapproimate valuesisoften a matter of judgment that dependson the purpose for which the valuesare to be used. Numerical approimationsoften provide a sense of size that eact values do not, but the can introduce severe errors if not used with care.
7 A68 Appendi G: Distance, Circles, and Quadratic Equations Eample From Figure G.9 we see that the parabola = has no intercepts. This can also be seen algebraicall b solving for the intercepts. Setting = 0 and solving the resulting equation s b the quadratic formula ields = 0 = 4 ± 6 0 = ± i 0 Because the solutions are not real numbers, there are no intercepts. Distance (m) Earth surface s Time (s) Figure G.0 t Eample A ball is thrown straight up from the surface of the Earth at time t = 0s with an initial velocit of 4.5 m/s. If air resistance is ignored, it can be shown that the distance s (in meters) of the ball above the ground after t seconds is given b s = 4.5t 4.9t (9) (a) Graph s versus t, making the tais horizontal and the sais vertical. (b) How high does the ball rise above the ground? Solution (a). Equation (9) is of form (7) with a = 4.9, b = 4.5, and c = 0, so b (8) the tcoordinate of the verte is t = b 4.5 = a ( 4.9) =.5 s and consequentl the scoordinate of the verte is b/(a) Verte The factored form of (9) is s = 4.5(.5) 4.9(.5) = m s = 4.9t(5 t) so the graph has tintercepts t = 0 and t = 5. From the verte and the intercepts we obtain the graph shown in Figure G.0. = a + b + c a > 0 Solution (b). From the scoordinate of the verte we deduce that the ball rises m above the ground. b/(a) Verte THE GRAPH of = a + b + c If and are interchanged in (7), the resulting equation, = a + b + c = a + b + c a < 0 is called a quadratic equation in. The graph of such an equation is a parabola with its line of smmetr parallel to the ais and its verte at the point with coordinate = b/(a) (Figure G.). Some problems relating to such equations appear in the eercises. Figure G.
8 Appendi G: Distance, Circles, and Quadratic Equations A69 EXERCISE SET G. Where in this section did we use the fact that the same scale was used on both coordinate aes? 5 Find (a) the distance between A and B (b) the midpoint of the line segment joining A and B.. A(, 5), B(, ) 3. A(7, ), B(, 9) 4. A(, 0), B( 3, 6) 5. A(, 6), B( 7, 4) 6 0 Use the distance formula to solve the given problem. 6. Prove that (, ), (, 8), and (4, 0) lie on a straight line. 7. Prove that the triangle with vertices (5, ), (6, 5), (, ) is isosceles. 8. Prove that (, 3), (4, ), and (, 6) are vertices of a right triangle and then specif the verte at which the right angle occurs. 9. Prove that (0, ), ( 4, 8), and (3, ) lie on a circle with center (, 3). 0. Prove that for all values of t the point (t, t 6) is equidistant from (0, 4) and (8, 0).. Find k, given that (,k)is equidistant from (3, 7) and (9, ).. Find and if (4, 5) is the midpoint of the line segment joining ( 3, ) and (, ). 3 4 Find an equation of the given line. 3. The line is the perpendicular bisector of the line segment joining (, 8) and ( 4, 6). 4. The line is the perpendicular bisector of the line segment joining (5, ) and (4, 8). 5. Find the point on the line = 0 that is equidistant from (3, 3) and (7, 3). [Hint: First find an equation of the line that is the perpendicular bisector of the line segment joining (3, 3) and (7, 3).] 6. Find the distance from the point (3, ) to the line (a) = 4 (b) =. 7. Find the distance from (, ) to the line = 0. [Hint: Find the foot of the perpendicular dropped from the point to the line.] 8. Find the distance from (8, 4) to the line = 0. [Hint: See the hint in Eercise 7.] 9. Use the method described in Eercise 7 to prove that the distance d from ( 0, 0 ) to the line A + B + C = 0is d = A 0 + B 0 + C A + B 0. Use the formula in Eercise 9 to solve Eercise 7.. Use the formula in Eercise 9 to solve Eercise 8.. Prove: For an triangle, the perpendicular bisectors of the sides meet at a point. [Hint: Position the triangle with one verte on the ais and the opposite side on the ais, so that the vertices are (0,a), (b, 0), and (c, 0).] 3 4 Find the center and radius of each circle. 3. (a) + = 5 (b) ( ) + ( 4) = 6 (c) ( + ) + ( + 3) = 5 (d) + ( + ) = 4. (a) + = 9 (b) ( 3) + ( 5) = 36 (c) ( + 4) + ( + ) = 8 (d) ( + ) + = 5 3 Find the standard equation of the circle satisfing the given conditions. 5. Center (3, ); radius = Center (, 0); diameter = Center ( 4, 8); circle is tangent to the ais. 8. Center (5, 8); circle is tangent to the ais. 9. Center ( 3, 4); circle passes through the origin. 30. Center (4, 5); circle passes through (, 3). 3. A diameter has endpoints (, 0) and (0, ). 3. A diameter has endpoints (6, ) and (, 3) Determine whether the equation represents a circle, a point, or no graph. If the equation represents a circle, find the center and radius = = = = = = = 40. ( /4) + ( /4) = = = = = Find an equation of (a) the bottom half of the circle + = 6 (b) the top half of the circle = Find an equation of (a) the right half of the circle + = 9 (b) the left half of the circle = Graph (a) = 5 (b) =
9 A70 Appendi G: Distance, Circles, and Quadratic Equations 48. Graph (a) = 4 (b) = Find an equation of the line that is tangent to the circle + = 5 at the point (3, 4) on the circle. 50. Find an equation of the line that is tangent to the circle at the point P on the circle (a) + + = 9; P(, ) (b) = 3; P(4, 3). 5. For the circle + = 0 and the point P(, ): (a) Is P inside, outside, or on the circle? (b) Find the largest and smallest distances between P and points on the circle. 5. Follow the directions of Eercise 5 for the circle + 4 = 0 and the point P ( ) 3, Referring to the accompaning figure, find the coordinates of the points T and T, where the lines L and L are tangent to the circle of radius with center at the origin. L L T T (3, 0) Figure E Apoint (, ) moves so that its distance to (, 0) is times its distance to (0, ). (a) Show that the point moves along a circle. (b) Find the center and radius. 55. A point (, ) moves so that the sum of the squares of its distances from (4, ) and (, 5) is 45. (a) Show that the point moves along a circle. (b) Find the center and radius. 56. Find all values of c for which the sstem of equations { = 0 ( c) + = has 0,,, 3, or 4 solutions. [Hint: Sketch a graph.] Graph the parabola and label the coordinates of the verte and the intersections with the coordinate aes. 57. = = = = = = = ( ) 64. = (3 + ) = = = = = = Find an equation of (a) the right half of the parabola = 3 (b) the left half of the parabola =. 7. Find an equation of (a) the upper half of the parabola = 5 (b) the lower half of the parabola =. 73. Graph (a) = + 5 (b) = Graph (a) = + 4 (b) = If a ball is thrown straight up with an initial velocit of 3 ft/s, then after t seconds the distance s above its starting height, in feet, is given b s = 3t 6t. (a) Graph this equation in a tscoordinate sstem (tais horizontal). (b) At what time t will the ball be at its highest point, and how high will it rise? 76. A rectangular field is to be enclosed with 500 ft of fencing along three sides and b a straight stream on the fourth side. Let be the length of each side perpendicular to the stream, and let be the length of the side parallel to the stream. (a) Epress in terms of. (b) Epress the area A of the field in terms of. (c) What is the largest area that can be enclosed? 77. A rectangular plot of land is to be enclosed using two kinds of fencing. Two opposite sides will have heavdut fencing costing $3/ft, and the other two sides will have standard fencing costing $/ft. A total of $600 is available for the fencing. Let be the length of each side with the heavdut fencing, and let be the length of each side with the standard fencing. (a) Epress in terms of. (b) Find a formula for the area A of the rectangular plot in terms of. (c) What is the largest area that can be enclosed? 78. (a) B completing the square, show that the quadratic equation = a + b + c can be rewritten as ( = a + b ) ) + (c b a 4a if a = 0. (b) Use the result in part (a) to show that the graph of the quadratic equation = a + b + c has its high point at = b/(a)if a<0 and its low point there if a> Solve the given inequalit. 79. (a) + 5 < 0 (b) + 3 > (a) + > 0 (b) < 0 8. At time t = 0 a ball is thrown straight up from a height of 5 ft above the ground. After t seconds its distance s, in feet, above the ground is given b s = t 6t. (a) Find the maimum height of the ball above the ground.
10 Appendi G: Distance, Circles, and Quadratic Equations A7 (b) Find, to the nearest tenth of a second, the time when the ball strikes the ground. (c) Find, to the nearest tenth of a second, how long the ball will be more than ft above the ground. 8. Find all values of at which points on the parabola = lie below the line = + 3.
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 informationSolving 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 informationTHE 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 information1. 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 information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More informationSECTION 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 informationREVIEW 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 information2.6. 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 informationD.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 informationIn 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 informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationax 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 informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree 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 informationThe 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 information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationINVESTIGATIONS 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 informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationEquation 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 informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationIdentifying 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 informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationSTRAND: 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 informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationMATH 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 informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationSECTION 22 Straight Lines
 Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationLinear 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 informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationGraphing 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 informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationThe Geometry of Piles of Salt Thinking Deeply About Simple Things
The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationSection 63 DoubleAngle and HalfAngle Identities
63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationREVIEW OF CONIC SECTIONS
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationSection 23 Quadratic Functions
118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationSECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xyplane. It is sketched in Figure 11.
SECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationSample Problems. Practice Problems
Lecture Notes Circles  Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
More informationWhen 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 informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationMath 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 informationPolynomial and Rational Functions
Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and
More informationAnswers (Anticipation Guide and Lesson 101)
Answers (Anticipation Guide and Lesson 0) Lesson 0 Copright Glencoe/McGrawHill, a division of The McGrawHill Companies, Inc. 0 NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get
More informationLesson 2: Circles, Chords, Diameters, and Their Relationships
Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information1.6. 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 informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationSection 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 informationLinear 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 information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationCCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:
GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M91.G..1 Prove that all circles are similar. M91.G.. Identify and describe relationships
More informationSLOPE 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 informationI 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 informationDownloaded 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