3.3. section. 140 (3-20) Chapter 3 Graphs and Functions in the Cartesian Coordinate System FIGURE FOR EXERCISE 52 MISCELLANEOUS

Transcription

1 0 (-0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Selling price (in thousands of dollars) 0 a) Use the graph on the net page to estimate the average retail price of a -ear-old car in 998. $,000 b) Find the slope of the line shown in the figure. 0 c) Use the slope to predict the price of a -ear-old car. $, (,,9) (,,09) Age (in ears) FIGURE FOR EXERCISE MISCELLANEOUS. The points (, ) and (,7) are on the line that passes through (, ) and has slope. Find the missing coordinates of the points. (, ), (0, 7). If a line passes through (, ) and has slope, then what is the value of on this line when 8,, and?, 6, 6. Find k so that the line through (, k) and (, ) has slope. 6. Find k so that the line through (k,)and(, 0) has slope. or 7. What is the slope of a line that is perpendicular to a line with slope 0.7? What is the slope of a line that is perpendicular to the line through (.7,.6) and (.8,.6)?.76 GETTING MORE INVOLVED 9. Writing. What is the difference between zero slope and undefined slope? A horizontal line has a zero slope and a vertical line has undefined slope. 60. Writing. Is it possible for a line to be in onl one quadrant? Two quadrants? Write a rule for determining whether a line has positive, negative, zero, or undefined slope from knowing in which quadrants the line is found. Ever line goes through at least two quadrants. A nonhorizontal, nonvertical line that misses quadrant II or IV or both has a positive slope. A nonhorizontal, nonvertical line that misses quadrant I or III or both has a negative slope. 6. Eploration. A rhombus is a quadrilateral with four equal sides. Draw a rhombus with vertices (, ), (0, ), (, ), and (, ). Find the slopes of the diagonals of the rhombus. What can ou conclude about the diagonals of this rhombus?,, perpendicular 6. Eploration. Draw a square with vertices (, ), (, ), (, ), and (, ). Find the slopes of the diagonals of the square. What can ou conclude about the diagonals of this square?,, perpendicular GRAPHING CALCULATOR EXERCISES 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does increasing m have on the graph of the equation? What are the slopes of these four lines? Increasing m makes the graph increase faster. The slopes of these lines are,,, and. 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does decreasing m have on the graph of the equation? What are the slopes of these four lines? Decreasing m makes the graph decrease faster. The slopes of these lines are,,, and. In this section Point-Slope Form Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Linear Functions. THREE FORMS FOR THE EQUATION OF A LINE In Section. ou learned how to graph a straight line corresponding to a linear equation. The line contains all of the points that satisf the equation. In this section we start with a line or a description of a line and write an equation corresponding to the line. Point-Slope Form Figure.8 shows the line that has slope and contains the point (, ). In Section. ou learned that the slope is the same no matter which two points of the line

2 . Three Forms for the Equation of a Line (-) stud tip No two students learn in the same wa or at the same speed. No one can tell ou eactl how to stud and learn. Learning is personal.you must discover what it takes for ou to learn mathematics and then to do whatever it takes. are used to calculate it. So if we find the slope m for this line using an arbitrar (, ) 8 point of the line, sa (, ), and the specific point (, ), we get 7 6 (, ) m. Because the slope of this line is, we can write Multipling each side b, we get FIGURE.8 ( ). Because (, ) was an arbitrar point on the line, this equation is satisfied b ever point on the line. If we use (, ) as the specific point and (, ) as an arbitrar point on a line with slope m, we can write m. Multipling each side of this equation b gives us the point-slope form of the equation of the line. Point-Slope Form The equation of the line through (, ) with slope m in point-slope form is m( ). E X A M P L E Graph and check that the line goes through (, ) b using the TRACE feature. 0 calculator close-up Writing an equation for a line given a point and the slope Find an equation for the line through (, ) with slope and solve it for. Use,, and m in the point-slope form: Now solve the equation for : [ ()] [ ] 6 If ou know two points on a line, then ou can graph the line (two points determine a line). In the net eample we will see that two points of a line also determine an equation for the line.

3 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem E X A M P L E calculator close-up Graph and check that the line goes through (, ) and (, ) b using the TRACE feature. 0 Writing an equation for a line given two points on the line Find an equation for the line through (, ) and (, ) and solve it for. We are not given the slope, but we can find it because the points (, ) and (, ) are on the line: () m Now use this slope and one of the points, sa (, ), to write the equation in pointslope form: () ( ) Point-slope form 9 Distributive propert 0 0 Solve for : Note that we would get the same equation if we had used slope and the other point (, ). Tr it. For the net eample, recall that if a line has slope m, then the slope of an line perpendicular to it is m, provided that m 0. E X A M P L E stud tip When taking a test, put a check mark beside ever question that ou have answered and checked. When ou have finished the test, then ou can go back and spend the remaining time on the problems that are not et checked. You will not waste time reworking problems that ou know are correct. An equation of a line perpendicular to another line Line l goes through (, 0) and is perpendicular to the line through (, ) and (, ). Find the equation of line l and then solve it for. First find the slope of the line through (, ) and (, ): () m 6 Because line l is perpendicular to this line, line l has slope. Now use (, 0) and the slope in the point-slope formula to get the equation of line l: 0 ( ) Distributive propert calculator close-up With slope-intercept form and a graphing calculator, it is eas to see how the slope affects the steepness of a line. The graphs of,,, and are all shown on the accompaning screen

4 Slope-Intercept Form. Three Forms for the Equation of a Line (-) The line in Eample has slope. To find the -intercept of this line, let 0 in : (0). The -intercept is (0, ). Its -coordinate appears in the equation: Slope -intercept (0, ) Because the slope and -intercept can be read from the equation when it is solved for, this form of the equation of the line is called slope-intercept form. Slope-Intercept Form The equation of a line in slope-intercept form is m b, where m is the slope and (0, b) is the -intercept. E X A M P L E FIGURE.9 Writing an equation given its slope and -intercept Write the slope-intercept form of the equation of the line shown in Fig..9. From Fig..9 we see that the -intercept is (0, ). If we start at the -intercept and move down and to the right, we get to another point on the line. So the slope is. The equation of this line in slope-intercept form is. stud tip Standard Form If students paid $ each and adults paid $7 each to attend a pla for which the ticket sales totaled $900, then we can write the equation This form of a linear equation is common in applications. It is called standard form. Get in the habit of checking our work and having confidence in our answers.the answers to the odd-numbered eercises are in the back of this book, but ou should look in the answer section onl after ou have checked on our own. You will not alwas have an answer section available. Standard Form The equation of a line in standard form is A B C, where A, B, and C are real numbers with A and B not both zero. The numbers A, B, and C in standard form can be an real numbers, but it is a common practice to write standard form using onl integers and a positive coefficient for. E X A M P L E Changing to standard form Write the equation in standard form using onl integers and a positive coefficient for.

5 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem helpful hint Solve A B C for, to get A C B B. So the slope of A B C is A. This fact can be used in B checking standard form. The slope of in Eample is or, which is the slope of the original equation. E X A M P L E 6 helpful hint Note that ever term in a linear equation in two variables is either a constant or a multiple of a variable. That is wh equations in one variable of the form a b 0 were called linear equations in Chapter. E X A M P L E 7 Use the properties of equalit to get the equation in the form A B C: Original equation Subtract from each side. Multipl each side b to get integral coefficients. Distributive propert Multipl b to make the coefficient of positive. To find the slope and -intercept of a line written in standard form, we convert the equation to slope-intercept form. Changing to slope-intercept form Find the slope and -intercept of the line. Solve for to get slope-intercept form: Original equation Divide each side b. Subtract from each side. The slope is, and the -intercept is 0,. You learned in Section. that the graph of the equation is a vertical line. Because slope is undefined for vertical lines, the equation of this line cannot be written in slope-intercept form or point-slope form. Onl nonvertical lines can be written in those forms. However, a vertical line can be written in standard form. For eample, can be written as 0. Ever line has an equation in standard form. Finding the equation of a line Write an equation in standard form with integral coefficients for the line l through (, ) that is perpendicular to the line. First solve the equation for to find its slope: The slope is.

6 . Three Forms for the Equation of a Line (-) Graph = and to check that is perpendicular to and that goes through (, ).The lines will look perpendicular onl if the same unit length is used on both aes. calculator close-up 0 0 Some calculators have a feature that adjusts the window to get the same unit length on both aes. The slope of line l is the opposite of the reciprocal of. So line l has slope and goes through (, ). Now use the point-slope form to write the equation: ( ) Point-slope form Distributive propert Multipl each side b. So is the standard form of the equation of the line through (, ) that is perpendicular to. Using Slope-Intercept Form for Graphing In the slope-intercept form, a point on the line (the -intercept) and the slope are readil available. To graph a line, we can start at the -intercept and count off the rise and run to get a second point on the line. E X A M P L E 8 Rise Run = + -intercept FIGURE.0 Using slope and -intercept to graph Graph the line. First write the equation in slope-intercept form: The slope is, and the -intercept is (0, ). Start at (0, ) on the -ais, then rise and run to locate a second point on the line. Because there is onl one line containing an two given points, these two points determine the line. See Fig..0. The three methods that we used for graphing linear equations are summarized as follows. Methods for Graphing a Linear Equation. Arbitraril select some points that satisf the equation, and draw a line through them.. Find the - and -intercepts (provided that the are not the origin), and draw a line through them.. Start at the -intercept and use the slope to locate a second point, then draw a line through the two points.

7 6 (-6) Chapter Graphs and Functions in the Cartesian Coordinate Sstem If the -coordinate of the -intercept is an integer and the slope is a rational number, then it is usuall the easiest to use the -intercept and slope. Linear Functions The linear equation m b with m 0 is a formula that shows how to determine a value of from a value of. We sa that is a linear function of. Functions in general will be discussed in Section.. In the net eample we use the pointslope formula to write Fahrenheit temperature as a linear function of Celsius temperature. E X A M P L E 9 Writing a linear function given two points Fahrenheit temperature F is a linear function of Celsius temperature C. Water freezes at 0 C or F and boils at 00 C or F. Find the linear equation that epresses F as a linear function of C. We want the equation of the line that contains the points (0, ) and (00, ) as shown in Fig... Use C as the independent variable () and F as the dependent variable (). The slope of the line is F m F C C Degrees Fahrenheit F Water boils (00, ) Water freezes (0, ) Degrees Celsius C 9 FIGURE. Using a slope of and the point (00, ) in the point-slope formula, we get 9 F (C 00). We can solve this equation for F to get the familiar formula relating Celsius and Fahrenheit temperature: 9 F C Because we knew the intercept (0, ), we could have used it and the slope 9 in slope-intercept form to write F 9 C. WARM-UPS True or false? Eplain our answer.. There is eactl one line through a given point with a given slope. True. The line a m( b) goes through (a, b) and has slope m. False. The equation of the line through (a, b) with slope m is m b. False

8 . Three Forms for the Equation of a Line (-7) 7 WARM-UPS (continued). The -coordinate of the -intercept of a nonvertical line is 0. True. The -coordinate of the -intercept of a nonhorizontal line is 0. True 6. Ever line in the -plane has an equation in slope-intercept form. False 7. The line 7 has slope. True 8. The line is perpendicular to the line. False 9. The line has a -intercept of (0, ). False 0. Ever line in the -plane has an equation in standard form. True. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is point-slope form? Point-slope form is m( ), where m is the slope and (, ) is a point on the line.. What is slope-intercept form? Slope-intercept form is m b, where m is the slope and (0, b) is the -intercept.. What two bits of information must ou have to write the equation of a line from a description of the line? To write an equation of a line, we need the slope and a point on the line.. What is standard form? Standard form is A B C, where A, B, and C are real numbers with A and B not both zero.. How do ou find the slope of a line when its equation is given in standard form? To find the slope from standard form, solve the equation for to get the form m b, where m is the slope. 6. How do ou graph a line when its equation is given in slope-intercept form? To graph a line knowing the slope and -intercept, start at the -intercept and count off the rise and run to locate a second point. Then draw a line through the -intercept and our second point. Find the equation of line l in each case and solve it for. See Eamples. 7. Line l goes through (, ) and has slope Line l goes through (, ) and has slope Line l goes through (, ) and has slope. 0. Line l goes through (, ) and has slope.. Line l goes through (, ) and (, 6) Line l goes through (, ) and (, ).. Line l goes through (, ) and is perpendicular to the line through (, ) and (, ). 8. Line l goes through (0, 0) and is perpendicular to the line through (0, 6) and (, 0). 6. Line l goes through (0, 0) and is parallel to the line through (9, ) and (, 6). 6. Line l goes through (, ) and is parallel to the line through (6, ) and (, 6). In Eercises 7 6, write an equation in slope-intercept form (if possible) for each of the lines shown. See Eample. 7.

9 8 (-8) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Write each equation in standard form using onl integers and a positive coefficient for. See Eample ( ) 0. ( 6). ( ) 6. ( ) Write each equation in slope-intercept form, and identif the slope and -intercept. See Eample 6..,, 0,

10 . Three Forms for the Equation of a Line (-9) 9 6.,, 0, 7. 0,, (0, ) 8. 0,, 0, 9., 0, (0, ) , 0, (0, 9). ( ),, (0, ). ( ) 6,, (0, 6).,, (0, ). 6 6,, 0, 6. 7,, 0, 7 6.,, 0, ( 700) , 0.0, (0, 607) ( 990) , 0.0, (0, 900.) Find the equation of line l in each case and then write it in standard form with integral coefficients. See Eample Line l has slope and goes through (0, ) Line l has slope and goes through 0,. 0. Line l has -intercept (, 0) and -intercept (0, ).. Line l has -intercept (0, ) and -intercept (, 0). 0. Line l goes through (, ) and is parallel to 6.. Line l goes through (, ) and is parallel to. 0. Line l is parallel to and goes through (, ) Line l is parallel to 7 and goes through (, ). 7. Line l goes through (, ) and is perpendicular to. 8. Line l goes through (, ) and is perpendicular to Line l goes through (, ) and is perpendicular to Line l is perpendicular to 0 and goes through (, 7). 6. Line l goes through (, ) and is parallel to the -ais. 6. Line l goes through (, 6) and is parallel to the -ais. Graph each line. Use the slope and -intercept when possible. See Eample

11 0 (-0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Determine whether each pair of lines is parallel or perpendicular , 7 Perpendicular 80., Perpendicular 8. 9, 8 Parallel 8. 6, Parallel , Perpendicular 8. 9, 8 Parallel Solve each problem. See Eample Heating water. Suppose the temperature, t, of a cup of water is a linear function of the number of seconds, s, that it is in the microwave. If the temperature at s 0 second is t 60 F and the temperature at s 0 seconds is 00 F, find the linear equation that epresses t as a function of s. What should the temperature be after 0 seconds? (Hint: Write the equation of the line containing the points (0, 60) and (0, 00) in the form t ms b.) Draw a graph of this linear function. t 7 s 60, 9F Making circuit boards. The accountant at Apollo Manufacturing has determined that the cost, C, per week in dollars for making circuit boards is a linear function of the number, n, of circuit boards produced in a week. If C $00 when n 000, and C $000 when n 000, find the linear equation that epresses C in terms of n. What is the cost if Apollo produces onl one circuit board in a week? Draw a graph of this linear function. C n 000, $ Carbon dioide emission. Worldwide emission of carbon dioide (CO ) increased from billion tons in 970 to billion tons in 99 (World Resources Institute, a) Find the equation of the line through (970, ) and (99, ) b) Use the equation to predict the worldwide emission of CO in billion tons 88. World energ use. Worldwide energ use in all forms increased from the equivalent of. billion tons of oil in 970 to the equivalent of 6 billion tons of oil in 99 (World Resources Institute, a) Find the equation of the line through (70,.) and (9, 6). 0.. b) Use the equation to predict the worldwide energ use in billion tons 89. Depth and flow. On Ma, 998 the depth of the water in the Tangipahoa River at Robert, Louisiana was 8. feet and the flow was 0. cubic feet per second (ft /sec). On Ma 8 the depth was 7.6 feet and the flow was 77. cubic feet per second (U.S. Geological Surve, Water Resources Data for Louisiana, 998). The flow w is a linear function of the depth d. Flow (thousands of ft /sec) 0 0 Depth (feet) FIGURE FOR EXERCISE 89

12 . Linear Inequalities and Their Graphs (-) a) Write the equation of the line through (8., 0.) and (7.6, 77.) and epress w as a linear function of d. b) What is the flow when the depth is 7.8 feet? c) Is the flow increasing or decreasing as the depth increases? a) w 0.d 9. b) 88.6 ft /sec c) increasing 90. Buing stock. On Jul, 998 a mutual fund manager spent $,0,0 on shares of Ford Motor Stock at $8. per share and shares of General Motors stock at $7.0 per share. a) Write a linear equation that models this situation. b) If,000 shares of Ford were purchased, then how man shares of GM were purchased? c) What are the intercepts of the graph of the linear equation? Interpret the intercepts. d) As the number of shares of Ford increases, does the number of shares of GM increase or decrease? a) ,0,0 b) 6,000 c) (0, 0,9.), (86,7., 0), The intercepts give the number of shares if all of the mone was spent on onl one tpe of stock. d) decrease GM shares (in thousands) Ford shares (in thousands) FIGURE FOR EXERCISE 90 GETTING MORE INVOLVED 9. Eploration. Plot the points (, ), (, ), (, ), (, 6), and (, 7) on graph paper. Use a ruler to draw a straight line that best fits the five points. The line drawn does not necessaril have to go through an of the five points. a) Estimate the slope and -intercept for the line drawn and write an equation for the line in slope-intercept form. b) For each -coordinate from through, find the difference between the given -coordinate and the - coordinate on our line. c) To determine how well ou have done, square each difference that ou found in part (b) and then find the sum of those squares. Compare our sum with our classmates sums. The person with the smallest sum has done the best job of fitting a line to the five given points. GRAPHING CALCULATOR EXERCISES 9. Graph the equation 0. using the standard viewing window. Adjust the range of -values so that the line goes from the lower left corner of our viewing window to the upper right corner. 9. Graph 000, using a viewing window that shows both the -intercept and the -intercept. 9. Graph 00 and 0. on the same screen, using the viewing window and Should these lines be perpendicular? Eplain. The lines are perpendicular and will appear so in a window in which the length of one unit on the -ais is equal to the length of one unit on the -ais. 9. The lines and.9 are not parallel. Find a viewing window in which the lines intersect. Estimate the point of intersection. The lines intersect at (0, 97). In this section Definition Graphing Linear Inequalities The Test Point Method Graphing Compound Inequalities Applications. LINEAR INEQUALITIES AND THEIR GRAPHS In the first three sections of this chapter ou studied linear equations. We now turn our attention to linear inequalities. Definition A linear inequalit is a linear equation with the equal sign replaced b an inequalit smbol. Linear Inequalit If A, B, and C are real numbers with A and B not both zero, then A B C is called a linear inequalit. In place of, we can also use,, or.