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

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

Objectives Determine whether a function is linear. Graph a linear function given two points, a table, an equation, or a point and a slope.

Vocabulary linear function slope y-intercept x-intercept slope-intercept form

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.

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

Functions with a constant rate of change are called linear functions 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).

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

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

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

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

The constant rate of change for a linear function is its slope 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.

Example 2A: Graphing Lines Using Slope and a Point Graph the line with slope that passes 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.

Example 2B: Graphing Lines Using Slope and a Point Graph the line with slope that passes 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.

Check It Out! Example 2 Graph the line with slope that passes 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.

Recall from geometry that two points determine a line 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.

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

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

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.

Example 4A: Graphing Functions in Slope-Intercept Form Write the function –4x + y = –1 in slope-intercept form. Then graph the function. Solve for y first. –4x + y = –1 +4x +4x Add 4x to both sides. y = 4x – 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.

Example 4A 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.

Example 4B: 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.

Check It Out! Example 4A Write the function 2x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2x – y = 9 –2x –2x Add –2x to both sides. –y = –2x + 9 y = 2x – 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.

Check It Out! Example 4A 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.

Check It Out! Example 4B Write the function 5x = 15y + 30 in slope-intercept form. Then graph the function. Solve for y first. 5x = 15y + 30 –30 –30 Subtract 30 from both sides. 5x – 30 = 15y 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.

An equation with only one variable can be represented by either a vertical or a horizontal line.

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.

The slope of a vertical line is undefined. The slope of a horizontal line is zero.

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.) x = 2 y = –4 B. y = –4 This is a horizontal line located at the y-value –4.

Determine if each line is vertical or horizontal. 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 y = –5 B. x = 0.5 This is a vertical line located at the x-value 0.5.

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 Graph the line. Step 1 Find the slope. The y-intercept is the original altitude, 1800 ft. Use (0, 1800) and (2000, 3000) as two points on the line. Select a scale for each axis that will fit the data, and graph the function. The rise is 3000 – 1800, or 1200 ft. The run is 2000 ft. The slope is .

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 Find the average speed. Step 2 Graph the line. The y-intercept is the distance traveled at 0 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. distance = rate  time 192 mi = rate  3 h The slope is 64 mi/h.

Lesson Quiz: Part 1 1. Determine whether the data could represent a linear function. x –1 2 5 8 f(x) –3 1 9 yes 2. For 3x – 4y = 24, find the intercepts, write in slope- intercept form, and graph. x-intercept: 8; y-intercept: –6; y = 0.75x – 6

Lesson Quiz: Part 2 3. Determine if the line y = -3 is vertical or horizontal. 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