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Lesson 15 Representing Linear Functions TAKS Grade 9 Objective 3 (A.5)(C) Linear functions can be represented by equations, tables, graphs, and descriptions. New Vocabulary • slope-intercept form • slope • y-intercept Translating from Algebra to Charts and Graphs To translate an equation into a table or a graph, generate a set of points that satisfy the equation. A newspaper advertisement costs $4, plus $5 for each line of text. Write an equation for the cost of an advertisement. Then evaluate the equation to find the cost of a 1-line ad, a 2-line ad, a 3-line ad, and a 4-line ad. Model the equation with a table, and then graph your results. Step 1 Write an equation. Choose variables and relate them according to the problem situation. Let y represent the cost of an advertisement, and x represent the number of lines of text in the ad. The cost is equal to $4, plus $5 times the number of lines. y 4 5x Step 2 Make a table of values that shows x 4 5x y (x, y) the cost y for the given numbers 1 4 5(1) 4 5 9 (1, 9) of lines, x. 2 4 5(2) 4 10 14 (2, 14) 3 4 5(3) 4 15 19 (3, 19) 4 4 5(4) 4 20 24 (4, 24) Step 3 Graph your results. Use the ordered pairs in the table to sketch the graph. The points should not be connected, because you cannot buy fractions of a line. y Cost of Advertisement Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. EXAMPLE 1 20 15 10 5 O 2 4 6 8 x Number of Lines of Text Quick Check 1 1a. Model 8 4y 2x using a table of values and a graph. 1b. Carol painted black and white tiles for the floor in her kitchen. The number of white tiles was 32 plus 45 the number of black tiles. Write an equation for the number of white tiles in terms of the number of black ones. Model the equation with a table and graph your results. TAKS Review and Preparation Workbook LESSON 15 ■ Representing Linear Functions 43 TAKS Objective 3 (A.5)(C) LESSON 15 Translating into Algebra The slope-intercept form of a linear function relates y to x using the slope and the y-intercept. The slope of a line is the ratio of the vertical change to the horizontal change. To find the slope, divide the vertical change between any two points (x1, y1) and (x2, y2) by the horizontal change between them: y 2 y slope x22 2 x11. The y-intercept is the y-coordinate of the point where a line crosses the y-axis. A line with positive slope rises as it goes to the right. A line with negative slope falls as it goes to the right. To determine the equation of a line, find the slope and the y-intercept. Then write the equation in slope-intercept form, y mx b, where m represents the slope, and b represents the y-intercept. EXAMPLE 2 A group of students made this graph for their class. Write an equation that could represent this function. Step 1 Find the slope and use it to replace the y 4 m in the equation. Two points on the graph are (4, 1) and 2 (4, 5). Use the definition of slope: –4 y 2y m x22 2 x11 4x –2 O –2 21 25 24 2 4 –4 34 Step 2 Find the y-intercept and use that number to replace b. The graph crosses the y-axis at the point (0, 2). Thus, the y-intercept is 2. The equation is: y 34x 2. Quick Check 2 2a. Find the equation of the function shown in the following graph. 4 2b. Find the equation for the line passing through the ordered pairs shown. y y (3, 9) 8 2 6 –4 –2 O 2 4x 4 –2 –4 (–1, 1) –4 44 LESSON 15 ■ Representing Linear Functions –2 2 O 2 4 x TAKS Review and Preparation Workbook Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. 26 28 Name__________________________Class____________Date________ 1 Which function includes the data set {(3, 6), (4, 8), (8, 16)}? 4 Which points are included in the function y x 8? A yx3 F (0, 8), (1, 9), (1, 7) B y x2 G (2, 6), (3, 11), (1, 7) C yx6 J (2, 10), (4, 12), (5, 13) H (3, 5), (4, 4), (9, 1) D y 2x 5 Which graph represents the function y 15x 3? 2 What is the equation of the line shown? 4 4 y 2 A 2 –4 –2 O y 2 –4 –2 O 2 4x 2 4x 2 4x 2 4x –2 4x –4 –2 –4 4 Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. F y 12x 2 G y 21x 2 y 2 B –4 –2 O –2 H y 2x 2 –4 J y 2x 2 4 3 Rebecca and Carlos sold posters for a school fundraiser. Rebecca sold four more than half the number of posters that Carlos sold. Which equation could represent the number of posters Rebecca sold in terms of the number Carlos sold? 2 C –4 –2 O –2 –4 A r 12c 4 4 B r 2c 8 C r 2c 8 y y 2 D –4 –2 O –2 D r 21c 4 –4 TAKS Review and Preparation Workbook LESSON 15 ■ Representing Linear Functions 45