SECTION 22 Straight Lines


 Randolph Boone
 2 years ago
 Views:
Transcription
1  Straight Lines 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 the ends, what is the radius of the circle containing the arc? (B) If the compan decides to position the tower on this circle at a point directl east of town A, how far from town A should the place the tower? Compute answer to one decimal place. Rivet 9. Construction. Town B is located 6 miles east and 1 miles north of town A (see the figure). A local telephone compan wants to position a rela tower so that the distance from the tower to town B is twice the distance from the tower to town A. (A) Show that the tower must lie on a circle, find the center and radius of this circle, and graph. Tower Town A (, ) 96. Construction. Repeat Problem 9 if the distance from the tower to town A is twice the distance from the tower to town B. Town B (6, 1) SECTION  Straight Lines Graphs of FirstDegree Equations in Two Variables Slope of a Line Equations of a Line Special Forms Parallel and Perpendicular Lines In this section we will consider one of the most basic geometric figures a straight line. We will learn how to graph straight lines, given various standard equations, and how to find the equation of a straight line, given information about the line. Adding these important tools to our mathematical toolbo will enable us to use straight lines as an effective problemsolving tool, as evidenced b the application eercises at the end of this section. Graphs of First Degree Equations in Two Variables With our past eperience in graphing equations in two variables, ou probabl remember that firstdegree equations in two variables, such as 4 9 have graphs that are straight lines. This fact is stated in Theorem 1. For a partial proof of this theorem, see Problem 80 of the eercises at the end of this section. Theorem 1 The Equation of a Straight Line If A, B, and C are constants, with A and B not both 0, and and are variables, then the graph of the equation A B C Standard Form (1) is a straight line. An straight line in a rectangular coordinate sstem has an equation of this form.
2 116 Graphs and Functions Also, the graph of an equation of the form m b () where m and b are constants, is a straight line. Form (), which we will discuss in detail later, is simpl a special case of form (1) for B 0. This can be seen b solving form (1) for in terms of : A B C B B 0 To graph either equation (1) or (), we plot an two points from the solution set and use a straightedge to draw a line through these two points. The points where the line crosses the aes are convenient to use and eas to find. The intercept* is the ordinate of the point where the graph crosses the ais, and the intercept is the abscissa of the point where the graph crosses the ais. To find the intercept, let 0 and solve for ; to find the intercept, let 0 and solve for. It is often advisable to find a third point as a checkpoint. All three points must lie on the same straight line or a mistake has been made. EXAMPLE 1 Using Intercepts to Graph a Straight Line Graph the equation 4 1. Solution Find intercepts, a third checkpoint (optional), and draw a line through the two (three) points (Fig. 1). FIGURE intercept is (4, 0) (8, ) intercept is 4 (0, ) Check point 10 Matched Problem 1 Graph the equation 4 1. *If the intercept is a and the intercept is b, then the graph of the line passes through the points (a, 0) and (0, b). It is common practice to refer to both the numbers a and b and the points (a, 0) and (0, b) as the and intercepts of the line.
3  Straight Lines 117 To check the answer to Eample 1 on a graphing utilit, we first solve the equation for and then graph (Fig. ): FIGURE 10 Slope of a Line If we take two points P 1 ( 1, 1 ) and P (, ) on a line, then the ratio of the change in to the change in as we move from point P 1 to point P is called the slope of the line. Roughl speaking, slope is a measure of the steepness of a line. Sometimes the change in is called the run and the change in is called the rise. DEFINITION 1 Slope of a Line If a line passes through two distinct points P 1 ( 1, 1 ) and P (, ), then its slope m is given b the formula m P (, ) Vertical change (rise) Horizontal change (run) P 1 ( 1, 1 ) 1 Run 1 Rise (, 1 ) For a horizontal line, doesn t change as changes; hence, its slope is 0. On the other hand, for a vertical line, doesn t change as changes; hence, 1 and its slope is not defined: For a vertical line, slope is not defined. In general, the slope of a line ma be positive, negative, 0, or not defined. Each of these cases is interpreted geometricall as shown in Table 1.
4 118 Graphs and Functions TABLE 1 Geometric Interpretation of Slope Line Slope Eample Rising as moves from left to right Positive Falling as moves from left to right Negative Horizontal 0 Vertical Not defined In using the formula to find the slope of the line through two points, it doesn t matter which point is labeled P 1 or P, since changing the labeling will change the sign in both the numerator and denominator of the slope formula: For eample: 7 7 In addition, it is important to note that the definition of slope doesn t depend on the two points chosen on the line as long as the are distinct. This follows from the fact that the ratios of corresponding sides of similar triangles are equal. EXAMPLE Finding Slopes Sketch a line through each pair of points and find the slope of each line. (A) (, 4), (, ) (B) (, ), (1, ) (C) ( 4, ), (, ) (D) (, 4), (, ) Solutions (A) (B) (, ) (, ) (, 4) (1, ) m ( 4) ( ) m 1 ( ) 6
5  Straight Lines 119 (C) (D) ( 4, ) (, ) (, 4) (, ) m ( 4) m 4 ; 7 0 slope is not defined Matched Problem Find the slope of the line through each pair of points. Do not graph. (A) (, ), (, ) (B) (, 1), (1, ) (C) (0, 4), (, 4) (D) (, ), (, 1) Equations of a Line Special Forms Let us start b investigating wh m b is called the slope intercept form for a line. EXPLOREDISCUSS 1 (A) Graph b for b,, 0,, and simultaneousl in the same coordinate sstem. Verball describe the geometric significance of b. (B) Graph m 1 for m, 1, 0, 1, and simultaneousl in the same coordinate sstem. Verball describe the geometric significance of m. (C) Using a graphing utilit, eplore the graph of m b for different values of m and b. As ou can see from the above eploration, the constants m and b in m b () have special geometric significance, which we now eplicitl state. If we let 0, then b, and we observe that the graph of equation () crosses the ais at (0, b). The constant b is the intercept. For eample, the intercept of the graph of is. To determine the geometric significance of m, we proceed as follows: If m b, then b setting 0 and 1, we conclude that both (0, b) and (1, m b) lie on the graph, which is a line. Hence, the slope of this line is given b Slope 1 (m b) b m 1 1 0
6 10 Graphs and Functions Thus, m is the slope of the line given b m b. Now we know wh equation () is called the slope intercept form of an equation of a line. Theorem Slope Intercept Form m b m Rise Run Slope b intercept intercept b m b Rise Run EXAMPLE Using the Slope Intercept Form (A) Write the slope intercept equation of a line with slope and intercept. (B) Find the slope and intercept, and graph. 4 Solutions (A) Substitute m and b in m b to obtain b intercept FIGURE 4 (B) The intercept of 4 is, so the point (0, ) is on the graph. The slope of the line is 4, so when the coordinate of (0, ) increases (runs) b 4 units, the coordinate changes (rises) b. The resulting point (4, ) is easil plotted, and the two points ield the graph of the line. In short, we start at the intercept, and move 4 units to the right and units down to obtain a second point. We then draw a line through the two points, as shown in Figure. Matched Problem Write the slope intercept equation of the line with slope and intercept 1. Graph the equation. In Eample we found the equation of a line with a given slope and intercept. It is also possible to find the equation of a line passing through a given point with a given slope or to find the equation of a line containing two given points. Suppose a line has slope m and passes through a fied point ( 1, 1 ). If the point (, ) is an other point on the line, then 1 1 m 1
7  Straight Lines 11 that is, 1 m( 1 ) (4) We now observe that ( 1, 1 ) also satisfies equation (4) and conclude that (4) is an equation of a line with slope m that passes through ( 1, 1 ). We have just obtained the point slope form of the equation of a line. Theorem Point Slope Form An equation of a line through a point P 1 ( 1, 1 ) with slope m is 1 m( 1 ) Remember that P(, ) is a variable point and P 1 ( 1, 1 ) is fied. P(, ) P 1 ( 1, 1 ) The point slope form is etremel useful, since it enables us to find an equation for a line if we know its slope and the coordinates of a point on the line or if we know the coordinates of two points on the line. In the latter case, we find the slope first using the coordinates of the two points; then we use the point slope form with either of the two given points. EXAMPLE 4 Using the Point Slope Form (A) Find an equation for the line that has slope and passes through the point (, 1). Write the final answer in the standard form A B C. (B) Find an equation for the line that passes through the two points (4, 1) and ( 8, ). Write the final answer in the slope intercept form m b. Solutions (A) Let m and ( 1, 1 ) (, 1). Then 1 m( 1 ) 1 [ ( )] 1 ( ) 4 7 or 7
8 1 Graphs and Functions (B) First, find the slope of the line b using the slope formula: m 1 1 ( 1) Now let ( 1, 1 ) be either of the two given points and proceed as in part A we choose ( 1, 1 ) (4, 1): 1 m( 1 ) ( 1) 1 ( 4) 1 1 ( 4) You should verif that using ( 8, ), the other given point, produces the same equation. Matched Problem 4 (A) Find an equation for the line that has slope and passes through the point (, ). Write the final answer in the standard form A B C. (B) Find an equation for the line that passes through the two points (, 1) and (7, ). Write the final answer in the slope intercept form m b. EXAMPLE Business Markup Polic A sporting goods store sells a fishing rod that cost $60 for $8 and a pair of crosscountr ski boots that cost $80 for $106. (A) If the markup polic of the store for items that cost more than $0 is assumed to be linear and is reflected in the pricing of these two items, write an equation that relates retail price R to cost C. (B) Use the equation to find the retail price for a pair of running shoes that cost $40. (C) Check with a graphing utilit. Solutions (A) If the retail price R is assumed to be linearl related to cost C, then we are looking for an equation whose graph passes through (C 1, R 1 ) (60, 8) and (C, R ) (80, 106). We find the slope, and then use the point slope form to find the equation. m R R C C R R 1 m(c C 1 ) R 8 1.(C 60) R 8 1.C 7 R 1.C 10
9  Straight Lines 1 10 (B) R 1.(40) 10 $8 (C) The check is shown in Figure FIGURE 4 0 Matched Problem The management of a compan that manufactures ballpoint pens estimates costs for running the compan to be $00 per da at zero output and $700 per da at an output of 1,000 pens. (A) Assuming total cost per da C is linearl related to total output per da, write an equation relating these two quantities. (B) What is the total cost per da for an output of,000 pens? The simplest equations of lines are those for horizontal and vertical lines. Consider the following two equations: 0 a or a () 0 b or b (6) In equation (), can be an number as long as a. Thus, the graph of a is a vertical line crossing the ais at (a, 0). In equation (6), can be an number as long as b. Thus, the graph of b is a horizontal line crossing the ais at (0, b). We summarize these results as follows: Theorem 4 Vertical and Horizontal Lines Equation Graph a (short for 0 a) Vertical line through (a, 0) (Slope is undefined.) b (short for 0 b) Horizontal line through (0, b) (Slope is 0.) a b b a
10 14 Graphs and Functions EXAMPLE 6 Graphing Horizontal and Vertical Lines Graph the line and the line. Solution Matched Problem 6 Graph the line 4 and the line. The various forms of the equation of a line that we have discussed are summarized in Table for convenient reference. TABLE Equations of a Line Standard form A B C A and B not both 0 Slope intercept form m b Slope: m; intercept: b Point slope form 1 m( 1 ) Slope: m; Point: ( 1, 1 ) Horizontal line b Slope: 0 Vertical line a Slope: Undefined EXPLOREDISCUSS Determine conditions on A, B, and C so that the linear equation A B C can be written in each of the following forms, and discuss the possible number of and intercepts in each case. 1. m b, m 0. b. a Parallel and Perpendicular Lines From geometr, we know that two vertical lines are parallel to each other and that a horizontal line and a vertical line are perpendicular to each other. How can we tell
11  Straight Lines 1 when two nonvertical lines are parallel or perpendicular to each other? Theorem, which we state without proof, provides a convenient test. Theorem Parallel and Perpendicular Lines Given two nonvertical lines L 1 and L with slopes m 1 and m, respectivel, then L 1 L if and onl if m 1 m L 1 L if and onl if m 1 m 1 The smbols and mean, respectivel, is parallel to and is perpendicular to. In the case of perpendicularit, the condition m 1 m 1 also can be written as m 1 m 1 or m 1 1 m Thus: Two nonvertical lines are perpendicular if and onl if their slopes are the negative reciprocals of each other. EXAMPLE 7 Parallel and Perpendicular Lines Given the line: L: and the point P(, ), find an equation of a line through P that is: (A) Parallel to L (B) Perpendicular to L Write the final answers in the slope intercept form m b. Solutions First, find the slope of L b writing in the equivalent slope intercept form m b: Thus, the slope of L is. The slope of a line parallel to L is the same,, and the slope of a line perpendicular to L is. We now can find the equations of the two lines in parts A and B using the point slope form.
12 16 Graphs and Functions (A) Parallel (m ): (B) Perpendicular (m ): 1 m( 1 ) 1 m( 1 ) ( ) ( ) 9 19 Matched Problem 7 Given the Line L: 4 and the point P(, ), find an equation of a line through P that is: (A) Parallel to L (B) Perpendicular to L Write the final answers in the slope intercept form m b. Answers to Matched Problems 1.. (A) m 0 (B) m 1 (C) m 4 (D) m is not defined (A) 4 (B) 1. (A) C (B) $, (A) 1 (B) 4
13  Straight Lines 17 EXERCISE  A. In Problems 1 6, use the graph of each line to find the intercept, intercept, and slope. Write the slopeintercept form of the equation of the line Graph each equation in Problems 7 0, and indicate the slope, if it eists. Check our graphs in Problems 7 0 b graphing each on a graphing utilit In Problems 1 4, find the equation of the line with the indicated slope and intercept. Write the final answer in the standard form A B C, A Slope 1; intercept 0. Slope 1; intercept 7. Slope ; intercept 4 4. Slope ; intercept 6
14 18 Graphs and Functions B In Problems 8, find the equation of the line passing through the given point with the given slope. Write the final answer in the slopeintercept form m b.. (0, ); m 6. (4, 0); m 7. (, 4); m 8. (, ); m In Problems 9 4, find the equation of the line passing through the two given points. Write the final answer in the slopeintercept form m b or in the form c. 9. (, ); (4, ) 0. ( 1, 4); (, ) 1. (, ); (, ). (0, ); (, ). ( 4, ); (0, ) 4. (, 4); (, 6) In Problems 46, write an equation of the line that contains the indicated point and meets the indicated condition(s). Write the final answer in the standard form A B C, A 0.. (, 1); parallel to 7 6. (, ); parallel to 4 7. (0, 4); parallel to 9 8. (, 0); parallel to (, ); parallel to ais 40. (, 1); parallel to ais 41. (4, ); perpendicular to 4 4. ( 1, ); perpendicular to 4. (, 0); perpendicular to (0, ); perpendicular to 1 4. (, ); perpendicular to ais 46. (1, 7); perpendicular to ais In Problems 47 0, classif the quadrilateral ABCD with the indicated vertices as a trapezoid, a parallelogram, a rectangle, or none of these. 47. A(, ); B(8, 7); C(10, 1); D( 4, 6) 48. A(, ); B(, 4); C(6, 10); D(4, 4) 49. A(0, ); B(4, 1); C(1, ); D(, ) 0. A( 6, ); B(, 7); C(, 4); D( 4, 1) 1. Find the equation of the perpendicular bisector of the line segment joining ( 4, ) and (, 4) b using the pointslope form of the equation of a line.. Solve Problem 1 b using the distance between two points formula, and compare the results. 4 Problems 8 are calculusrelated. Recall that a line tangent to a circle at a point is perpendicular to the radius drawn to that point (see the figure). Find the equation of the line tangent to the circle at the indicated point. Write the final answer in the standard form A B C, A 0. Graph the circle and the tangent line on the same coordinate sstem.., (, 4) , ( 8, 6). 0, (, ) 6. 80, ( 4, 8) 7. ( ) ( 4) 169, (8, 16) 8. ( ) ( 9) 89, ( 1, 6) C 9. (A) Graph the following equations in the same coordinate sstem: 6 6 (B) From our observations in part A, describe the famil of lines obtained b varing C in A B C while holding A and B fied. (C) Verif our conclusions in part B with a proof. 60. (A) Graph the following two equations in the same coordinate sstem: (B) Graph the following two equations in the same coordinate sstem: 1 1 (C) From our observations in parts A and B, describe the apparent relationship of the graphs of A B C and B A C. (D) Verif our conclusions in part C with a proof.
15  Straight Lines 19 Sketch the graphs of the equations in Problems Describe the relationship between the graphs of m b and m b. (See Problems 61 and 6.) 68. Describe the relationship between the graphs of m b and m b. (See Problems 6 and 64.) 69. Prove that if a line L has intercept (a, 0) and intercept (0, b), then the equation of L can be written in the intercept form In Problems 70 and 71, write the equation of the line with the indicated intercepts in the standard form A B C, A (, 0) and (0, ) 71. (, 0) and (0, 7) 7. Let P 1 ( 1, 1 ) P 1 ( 1, m 1 b) P (, ) P (, m b) P (, ) P (, m b) be three arbitrar points that satisf m b with 1. Show that P 1, P, and P are collinear; that is, the lie on the same line. [Hint: Use the distance formula and show that d(p 1, P ) d(p, P ) d(p 1, P ).] This proves that the graph of m b is a straight line. APPLICATIONS 7. Boiling Point of Water. At sea level, water boils when it reaches a temperature of 1 F. At higher altitudes, the atmospheric pressure is lower and so is the temperature at which water boils. The boiling point B in degrees Fahrenheit at an altitude of feet is given approimatel b (A) Complete Table 1. TABLE 1 B ,000 10,000 1,000 0,000,000 0,000 B a b 1 a, b 0 (B) Based on the information in the table, write a brief verbal description of the relationship between altitude and the boiling point of water. 74. Air Temperature. As dr air moves upward, it epands and cools. The air temperature A in degrees Celsius at an altitude of kilometers is given approimatel b (A) Complete Table. TABLE 0 A A 9 (B) Based on the information in the table, write a brief verbal description of the relationship between altitude and air temperature. 7. Car Rental. A car rental agenc computes dail rental charges for compact cars with the equation c 0. where c is the dail charge in dollars and is the dail mileage. Translate this algebraic statement into a verbal statement that can be used to eplain the dail charges to a customer. 76. Installation Charges. A telephone store computes charges for phone installation with the equation c where c is the installation charge in dollars and is the time in minutes spent performing the installation. Translate this algebraic statement into a verbal statement that can be used to eplain the installation charges to a customer. Merck & Co., Inc., is the world s largest pharmaceutical compan. Problems 77 and 78 refer to the data in Table, taken from the compan s 199 annual report. TABLE Sales Net income 1 4 Selected Financial Data (billion $) for Merck & Co., Inc $.9 $ $6. $ Sales Analsis. A mathematical model for Merck s sales is given b where 0 corresponds to $7.7 $ $8.6 $ $9.7 $.4
16 10 Graphs and Functions (A) Complete Table 4. Round values of to one decimal place. (B) Sketch the graph of and the sales data on the same aes. (C) Use the modeling equation to estimate the sales in 199. In 000. (D) Write a brief verbal description of the compan s sales from 1988 to Income Analsis. A mathematical model for Merck s income is given b where 0 corresponds to (A) Complete Table. Round values of to one decimal place. TABLE Net income TABLE 4 Sales (B) Sketch the graph of the modeling equation and the income data on the same aes. (C) Use the modeling equation to estimate the income in 199. In 000. (D) Write a brief verbal description of the compan s income from 1988 to Phsics. The two temperature scales Fahrenheit (F) and Celsius (C) are linearl related. It is known that water freezes at F or 0 C and boils at 1 F or 100 C. (A) Find a linear equation that epresses F in terms of C. (B) If a European famil sets its house thermostat at 0 C, what is the setting in degrees Fahrenheit? If the outside temperature in Milwaukee is 86 F, what is the temperature in degrees Celsius? (C) What is the slope of the graph of the linear equation found in part A? (The slope indicates the change in Fahrenheit degrees per unit change in Celsius degrees.) Phsics. Hooke s law states that the relationship between the stretch s of a spring and the weight w causing the stretch is linear (a principle upon which all spring scales are constructed). For a particular spring, a pound weight causes a stretch of inches, while with no weight the stretch of the spring is 0. (A) Find a linear equation that epresses s in terms of w. (B) What weight will cause a stretch of.6 inches? (C) What is the slope of the graph of the equation? (The slope indicates the amount of stretch per pound increase in weight.) 81. Business Depreciation. A cop machine was purchased b a law firm for $8,000 and is assumed to have a depreciated value of $0 after ears. The firm takes straightline depreciation over the ear period. (A) Find a linear equation that epresses value V in dollars in terms of time t in ears. (B) What is the depreciated value after ears? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 8. Business Markup Polic. A clothing store sells a shirt costing $0 for $ and a jacket costing $60 for $9. (A) If the markup polic of the store for items costing over $10 is assumed to be linear, write an equation that epresses retail price R in terms of cost C (wholesale price). (B) What does a store pa for a suit that retails for $40? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 8. Flight Conditions. In stable air, the air temperature drops about F for each 1,000foot rise in altitude. (A) If the temperature at sea level is 70 F and a commercial pilot reports a temperature of 0 F at 18,000 feet, write a linear equation that epresses temperature T in terms of altitude A (in thousands of feet). (B) How high is the aircraft if the temperature is 0 F? (C) What is the slope of the graph of the equation found in part A? Interpret verball. 84. Flight Navigation. An airspeed indicator on some aircraft is affected b the changes in atmospheric pressure at different altitudes. A pilot can estimate the true airspeed b observing the indicated airspeed and adding to it about % for ever 1,000 feet of altitude. (A) If a pilot maintains a constant reading of 00 miles per hour on the airspeed indicator as the aircraft climbs from sea level to an altitude of 10,000 feet, write a linear equation that epresses true airspeed T (miles per hour) in terms of altitude A (thousands of feet). (B) What would be the true airspeed of the aircraft at 6,00 feet? (C) What is the slope of the graph of the equation found in part A? Interpret verball.
17  Functions Oceanograph. After about 9 hours of a stead wind, the height of waves in the ocean is approimatel linearl related to the duration of time the wind has been blowing. During a storm with 0knot winds, the wave height after 9 hours was found to be feet, and after 4 hours it was 40 feet. (A) If t is time after the 0knot wind started to blow and h is the wave height in feet, write a linear equation that epresses height h in terms of time t. (B) How long will the wind have been blowing for the waves to be 0 feet high? Epress all calculated quantities to three significant digits. 86. Oceanograph. As a diver descends into the ocean, pressure increases linearl with depth. The pressure is 1 pounds per square inch on the surface and 0 pounds per square inch feet below the surface. (A) If p is the pressure in pounds per square inch and d is the depth below the surface in feet, write an equation that epresses p in terms of d. (B) How deep can a scuba diver go if the safe pressure for his equipment and eperience is 40 pounds per square inch? 87. Medicine. Cardiovascular research has shown that above the 10 cholesterol level, each 1% increase in cholesterol level increases coronar risk %. For a particular age group, the coronar risk at a 10 cholesterol level is found to be and at a level of 1 the risk is found to be (A) Find a linear equation that epresses risk R in terms of cholesterol level C. (B) What is the risk for a cholesterol level of 60? (C) What is the slope of the graph of the equation found in part A? Interpret verball. Epress all calculated quantities to three significant digits. 88. Demographics. The average number of persons per household in the United States has been shrinking steadil for as long as statistics have been kept and is approimatel linear with respect to time. In 1900, there were about 4.76 persons per household and in 1990, about.. (A) If N represents the average number of persons per household and t represents the number of ears since 1900, write a linear equation that epresses N in terms of t. (B) What is the predicted household size in the ear 000? Epress all calculated quantities to three significant digits. SECTION  Functions Definition of a Function Functions Defined b Equations Function Notation Application A Brief Histor of the Function Concept The idea of correspondence plas a central role in the formulation of the function concept. You have alread had eperiences with correspondences in everda life. For eample: To each person there corresponds an age. To each item in a store there corresponds a price. To each automobile there corresponds a license number. To each circle there corresponds an area. To each number there corresponds its cube. One of the most important aspects of an science (managerial, life, social, phsical, computer, etc.) is the establishment of correspondences among various tpes of phenomena. Once a correspondence is known, predictions can be made. A chemist can use a gas law to predict the pressure of an enclosed gas, given its temperature. An engineer can use a formula to predict the deflections of a beam subject to different loads. A computer scientist can use formulas to compare the efficienc of algorithms for sorting data stored in a computer. An economist would like to be able to predict interest rates, given the rate of change of the mone suppl. And so on.
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 informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
a p p e n d i f 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
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 information3.4 The PointSlope Form of a Line
Section 3.4 The PointSlope Form of a Line 293 3.4 The PointSlope Form of a Line In the last section, we developed the slopeintercept form of a line ( = m + b). The slopeintercept form of a line is
More informationD.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 informationD.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 informationQ (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 information3 Rectangular Coordinate System and Graphs
060_CH03_13154.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 informationQuadratic 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 informationYears 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 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationThe 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 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 informationC1: Coordinate geometry of straight lines
B_Chap0_0805.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 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 information2.1 Equations of Lines
Section 2.1 Equations of Lines 1 2.1 Equations of Lines The SlopeIntercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is
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 information1.2 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 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 information2.2 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 informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
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 informationLesson 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 information2.4 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 information3.1 Quadratic Functions
33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial
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 informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More information2.3 Writing Equations of Lines
. Writing Equations of Lines In this section ou will learn to use pointslope form to write an equation of a line use slopeintercept form to write an equation of a line graph linear equations using the
More information2.2 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 informationCoordinate 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 informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationSlopeIntercept Form and PointSlope Form
SlopeIntercept Form and PointSlope Form In this section we will be discussing SlopeIntercept Form and the PointSlope Form of a line. We will also discuss how to graph using the SlopeIntercept Form.
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 informationSlope. SAFETY A ladder truck uses a moveable ladder to reach upper levels of houses and buildings.
9 MAIN IDEA Find the slope of a line. New Vocabular Slope SAFETY A ladder truck uses a moveable ladder to reach upper levels of houses and buildings. 1. The rate of change of the slope Math nline glencoe.com
More information4 Writing Linear Functions
Writing Linear Functions.1 Writing Equations in SlopeIntercept Form. Writing Equations in PointSlope Form.3 Writing Equations in Standard Form. Writing Equations of Parallel and Perpendicular Lines.5
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
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 informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationSECTION 91 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 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 informationLinear and Quadratic Functions
Chapter Linear and Quadratic Functions. 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
More informationSection 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 information10.2. 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 doublenapped
More informationFilling 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 informationWords Algebra Graph. m 5 y 2 2 y 1. slope. Find slope in real life
. Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 6. Ke Vocabular slope parallel perpendicular
More informationMath 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationThe 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 informationTHIS CHAPTER INTRODUCES the Cartesian coordinate
87533_01_ch1_p001066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
More informationThe SlopeIntercept Form
7.1 The SlopeIntercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph
More informationDISTANCE, 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 informationLesson 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 informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationP1. 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 information2.3 Quadratic Functions
. Quadratic Functions 9. 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: the
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 informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationAlex 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 information13 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 informationReview 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 informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationLearning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations
Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.
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 informationSummary, Review, and Test
798 Chapter 7 Sstems of Equations and Inequalities Preview Eercises Eercises 37 39 will help ou prepare for the material covered in the first section of the net chapter. 37. Solve the sstem: + + 2z = 9
More informationFunctions and Graphs. Chapter 2
BLITMC0A.099_7 /0/0 :07 AM Page 7 Functions and Graphs Chapter T he cost of mailing a package depends on its weight. The probabilit that ou and another person in a room share the same birthda depends
More information1.5 Shifting, Reflecting, and Stretching Graphs
7_00.qd /7/0 0: AM Page 7. Shifting, Reflecting, and Stretching Graphs Section. Shifting, Reflecting, and Stretching Graphs 7 Summar of Graphs of Parent Functions One of the goals of this tet is to enable
More informationSLOPES AND EQUATIONS OF LINES CHAPTER
CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8 The Slope of a Line 8 The Equation of a Line 83 Midpoint of a Line Segment 84 The Slopes of Perpendicular Lines 85 Coordinate Proof 86 Concurrence of the
More informationAnalyzing 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 informationGRAPHS 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 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 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 informationSection 72 Ellipse. Definition of an Ellipse The following is a coordinatefree definition of an ellipse: DEFINITION
7 Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located
More informationTHE 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 informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationSection 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 informationSimplification 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 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 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 informationReview Exercises. Review Exercises 83
Review Eercises 83 Review Eercises 1.1 In Eercises 1 and, sketch the lines with the indicated slopes through the point on the same set of the coordinate aes. Slope 1. 1, 1 (a) (b) 0 (c) 1 (d) Undefined.,
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 informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
 NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationREVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95
REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets
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 informationSection 14.5 Directional derivatives and gradient vectors
Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive  and directions.
More informationALGEBRA. Generate points and plot graphs of functions
ALGEBRA Pupils should be taught to: Generate points and plot graphs of functions As outcomes, Year 7 pupils should, for eample: Use, read and write, spelling correctl: coordinates, coordinate pair/point,
More informationAlgebra 2 Honors: Quadratic Functions. Student Focus
Resources: SpringBoard Algebra Online Resources: Algebra Springboard Tet Algebra Honors: Quadratic Functions Semester 1, Unit : Activit 10 Unit Overview In this unit, students write the equations of quadratic
More informationMATH chapter 1 Name and Section Number (10 points) Each question below is 10 points unless otherwise noted. Find the slope of the line.
MATH 1  chapter 1 Name and Section Number ( points) Each question below is points unless otherwise noted. Find the slope of the line. 1) Solve the problem. 8) The change in a certain engineer's salar
More informationEssential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)
5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bedandbreakfast.
More informationTranslating Points. Subtract 2 from the ycoordinates
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 information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationSTRETCHING, 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 informationSection 22 Linear Equations and Inequalities
22 Linear Equations and Inequalities 92. Petroleum Consumption. Analyzing data from the United States Energy Department for the period between 1920 and 1960 reveals that petroleum consumption as a percentage
More information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 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 informationLinear Equations and Arithmetic Sequences
CONDENSED LESSON 3.1 Linear Equations and Arithmetic Sequences In this lesson ou will write eplicit formulas for arithmetic sequences write linear equations in intercept form You learned about recursive
More informationAppendix D: Variation
Appendi D: Variation Direct Variation There are two basic types of linear models. The more general model has a yintercept that is nonzero. y = m + b, b 0 The simpler model y = k has a yintercept that
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 information