2 3 Graphing Linear Functions Warm Up Lesson

2 -3 Graphing. Linear. Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra Mc. Dougal 2 Algebra 2

2 -3 Graphing Linear Functions Warm Up Solve each equation for y. 1. 7 x + 2 y = 6 2. y = – 2 x – 8 3. If 3 x = 4 y + 12, find y when x = 0. y = – 3 4. If a line passes through (– 5, 0) and (0, 2), then it passes through all but which quadrant. IV Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Objectives Determine whether a function is linear. Graph a linear function given two points, a table, an equation, or a point and a slope. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Vocabulary linear function slope y-intercept x-intercept slope-intercept form Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Meteorologists begin tracking a hurricane's distance from land when it is 350 miles off the coast of Florida and moving steadily inland. The meteorologists are interested in the rate at which the hurricane is approaching land. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions +1 Time (h) Distance from Land (mi) 0 1 +1 +1 2 3 4 350 325 300 275 250 – 25 This rate can be expressed as +1 – 25. Notice that the rate of change is constant. The hurricane moves 25 miles closer each hour. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions with a constant rate of change are called linear functions. A linear function can be written in the form f(x) = mx + b, where x is the independent variable and m and b are constants. The graph of a linear function is a straight line made up of all points that satisfy y = f(x). Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 1 A: Recognizing Linear Functions Determine whether the data set could represent a linear function. +2 +2 +2 x – 2 0 2 4 f(x) 2 1 0 – 1 – 1 The rate of change, So the data set is linear. Holt Mc. Dougal Algebra 2 – 1 , is constant .

2 -3 Graphing Linear Functions Example 1 B: Recognizing Linear Functions Determine whether the data set could represent a linear function. +1 +1 +1 x 2 3 4 5 f(x) 2 4 8 16 +2 The rate of change, +4 +8 , is not constant. 2 ≠ 4 ≠ 8. So the data set is not linear. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Check It Out! Example 1 A Determine whether the data set could represent a linear function. +7 +7 +7 x f(x) 4 11 18 – 6 – 15 – 24 – 33 – 9 The rate of change, So the data set is linear. Holt Mc. Dougal Algebra 2 25 – 9 , is constant .

2 -3 Graphing Linear Functions Check It Out! Example 1 B Determine whether the data set could represent a linear function. – 2 – 2 x 10 8 6 4 f(x) 7 5 1 – 7 – 4 The rate of change, – 4 – 8 , is not constant. . So the data set is not linear. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions The constant rate of change for a linear function is its slope. The slope of a linear function is the ratio , or . The slope of a line is the same between any two points on the line. You can graph lines by using the slope and a point. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 2 A: Graphing Lines Using Slope and a Point Graph the line with slope through (– 1, – 3). Plot the point (– 1, – 3). The slope indicates a rise of 5 and a run of 2. Move up 5 and right 2 to find another point. Then draw a line through the points. Holt Mc. Dougal Algebra 2 that passes

2 -3 Graphing Linear Functions Example 2 B: Graphing Lines Using Slope and a Point Graph the line with slope through (0, 2). Plot the point (0, 2). The negative slope can be viewed as You can move down 3 units and right 4 units, or move up 3 units and left 4 units. Holt Mc. Dougal Algebra 2 that passes

2 -3 Graphing Linear Functions Check It Out! Example 2 Graph the line with slope through (3, 1). Plot the point (3, 1). The slope indicates a rise of 4 and a run of 3. Move up 4 and right 3 to find another point. Then draw a line through the points. Holt Mc. Dougal Algebra 2 that passes

2 -3 Graphing Linear Functions Recall from geometry that two points determine a line. Often the easiest points to find are the points where a line crosses the axes. The y-intercept is the y-coordinate of a point where the line crosses the x-axis. The x-intercept is the x-coordinate of a point where the line crosses the y-axis. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 3: Graphing Lines Using the Intercepts Find the intercepts of 4 x – 2 y = 16, and graph the line. Find the x-intercept: 4 x – 2 y = 16 4 x – 2(0) = 16 Substitute 0 for y. 4 x = 16 x=4 x-intercept The x-intercept is 4. Find the y-intercept: 4 x – 2 y = 16 4(0) – 2 y = 16 Substitute 0 for x. – 2 y = 16 y = – 8 The y-intercept is – 8. Holt Mc. Dougal Algebra 2 y-intercept

2 -3 Graphing Linear Functions Check It Out! Example 3 Find the intercepts of 6 x – 2 y = – 24, and graph the line. Find the x-intercept: 6 x – 2 y = – 24 6 x – 2(0) = – 24 Substitute 0 for y. 6 x = – 24 x = – 4 The x-intercept is – 4. y-intercept Find the y-intercept: 6 x – 2 y = – 24 6(0) – 2 y = – 24 Substitute 0 for x. x-intercept – 2 y = – 24 y = 12 The y-intercept is 12. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Linear functions can also be expressed as linear equations of the form y = mx + b. When a linear function is written in the form y = mx + b, the function is said to be in slope-intercept form because m is the slope of the graph and b is the y-intercept. Notice that slope-intercept form is the equation solved for y. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 4 A: Graphing Functions in Slope-Intercept Form Write the function – 4 x + y = – 1 in slope-intercept form. Then graph the function. Solve for y first. – 4 x + y = – 1 +4 x Add 4 x to both sides. y = 4 x – 1 The line has y-intercept – 1 and slope 4, which is. Plot the point (0, – 1). Then move up 4 and right 1 to find other points. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 4 A Continued You can also use a graphing calculator to graph. Choose the standard square window to make your graph look like it would on a regular grid. Press ZOOM, choose 6: ZStandard, press ZOOM again, and then choose 5: ZSquare. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 4 B: Graphing Functions in Slope-Intercept Form Write the function in slope-intercept form. Then graph the function. Solve for y first. Multiply both sides by Distribute. The line has y-intercept 8 and slope. Plot the point (0, 8). Then move down 4 and right 3 to find other points. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Check It Out! Example 4 A Write the function 2 x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2 x – y = 9 Add – 2 x to both sides. – 2 x –y = – 2 x + 9 y = 2 x – 9 Multiply both sides by – 1. The line has y-intercept – 9 and slope 2, which is. Plot the point (0, – 9). Then move up 2 and right 1 to find other points. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Check It Out! Example 4 A Continued You can also use a graphing calculator to graph. Choose the standard square window to make your graph look like it would on a regular grid. Press ZOOM, choose 6: ZStandard, press ZOOM again, and then choose 5: ZSquare. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Check It Out! Example 4 B Write the function 5 x = 15 y + 30 in slopeintercept form. Then graph the function. Solve for y first. 5 x = 15 y + 30 – 30 Subtract 30 from both sides. 5 x – 30 = 15 y Divide both sides by 15. The line has y-intercept – 2 and slope. Plot the point (0, – 2). Then move up 1 and right 3 to find other points. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions An equation with only one variable can be represented by either a vertical or a horizontal line. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Vertical and Horizontal Lines Vertical Lines Horizontal Lines The line x = a is a vertical line at a. The line y = b is a vertical line at b. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions The slope of a vertical line is undefined. The slope of a horizontal line is zero. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Example 5: Graphing Vertical and Horizontal Lines Determine if each line is vertical or horizontal. A. x = 2 This is a vertical line located at the x-value 2. (Note that it is not a function. ) B. y = – 4 This is a horizontal line located at the y-value – 4. Holt Mc. Dougal Algebra 2 x=2 y = – 4

2 -3 Graphing Linear Functions Check It Out! Example 5 Determine if each line is vertical or horizontal. A. y = – 5 This is a horizontal line located at the y-value – 5. x = 0. 5 B. x = 0. 5 This is a vertical line located at the x-value 0. 5. Holt Mc. Dougal Algebra 2 y = – 5

2 -3 Graphing Linear Functions Example 6: Application A ski lift carries skiers from an altitude of 1800 feet to an altitude of 3000 feet over a horizontal distance of 2000 feet. Find the average slope of this part of the mountain. Graph the elevation against the distance. Step 2 1 Graph Find the slope. line. The y-intercept rise is 3000 is– the 1800, original 1800 ft. or 1200 altitude, ft. Use (0, 1800) and (2000, The run 2000 ft. on the 3000) asistwo points line. Select a scale for each The slope is. axis that will fit the data, and graph the function. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Check It Out! Example 6 A truck driver is at mile marker 624 on Interstate 10. After 3 hours, the driver reaches mile marker 432. Find his average speed. Graph his location on I-10 in terms of mile markers. Step 1 2 Find Graph the average line. speed. The y-intercept the distance = rate is time distance traveled at 0 192 mi = rate 3 h hours, 0 ft. Use (0, 0) and (3, 192) as two points on the line. Select a scale for each axis that will fit the data, and graph the function. The slope is 64 mi/h. Holt Mc. Dougal Algebra 2

2 -3 Graphing Linear Functions Lesson Quiz: Part 1 1. Determine whether the data could represent a linear function. x – 1 2 5 8 f(x) – 3 1 5 9 2. For 3 x – 4 y = 24, find the intercepts, write in slopeintercept form, and graph. x-intercept: 8; y -intercept: – 6; y = 0. 75 x – 6 Holt Mc. Dougal Algebra 2 yes

2 -3 Graphing Linear Functions Lesson Quiz: Part 2 3. Determine if the line y = -3 is vertical or horizontal 4. The bottom edge of a roof is 62 ft above the ground. If the roof rises to 125 ft above ground over a horizontal distance of 7. 5 yd, what is the slope of the roof? 2. 8 Holt Mc. Dougal Algebra 2