UNIT 2.3 LINEAR FUNCTIONS UNIT 2.3 LINEAR FUNCTIONS

AND SLOPE-INTERCEPT FORMAND SLOPE-INTERCEPT FORM

Warm UpSolve 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

Determine whether a function is linear.

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

Objectives

linear functionslopey-interceptx-interceptslope-intercept form

Vocabulary

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.

Time (h) 0 1 2 3 4

Distance from Land (mi) 350 325 300 275 250

+1

–25

+1

–25

+1

–25

+1

–25

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. 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).

Determine whether the data set could represent a linear function.

Check It Out! Example 1A

x 4 11 18 25

f(x) –6 –15 –24 –33

+7

–9

+7

–9

+7

–9

The rate of change, , is constant . So

the data set is linear.

Determine whether the data set could represent a linear function.

Check It Out! Example 1B

x 10 8 6 4

f(x) 7 5 1 –7

–2

–4

–2

–4

–2

–8

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 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.

Check It Out! Example 2

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.

Graph the line with slope that passes through (3, 1).

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.

Check It Out! Example 3

Find the x-intercept: 6x – 2y = –24

Find the intercepts of 6x – 2y = –24, and graph the line.

6x – 2(0) = –24

Find the y-intercept: 6x – 2y = –24

6x = –24

x = –4

6(0) – 2y = –24

–2y = –24

y = 12

Substitute 0 for y.

The x-intercept is –4.

Substitute 0 for x.

The y-intercept is 12.

x-intercept

y-intercept

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.

Check It Out! Example 4A

Solve for y first.

Write the function 2x – y = 9 in slope-intercept form. Then graph the function.

2x – y = 9

–y = –2x + 9

Add –2x to both sides. –2x –2x

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

Solve for y first.

Write the function 5x = 15y + 30 in slope-intercept form. Then graph the function.

5x = 15y + 30

5x – 30 = 15y

Subtract 30 from both sides. –30 –30

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.

Check It Out! Example 5

Determine if each line is vertical or horizontal.

A. y = –5

B. x = 0.5

This is a horizontal line located at the y-value –5.

This is a vertical line located at the x-value 0.5.

x = 0.5

y = –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 1 Find the slope.

The rise is 3000 – 1800, or 1200 ft.

The run is 2000 ft.

The slope is .

Step 2 Graph the line.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.

Check It Out! Example 6A 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.

distance = rate time

The slope is 64 mi/h.

192 mi = rate 3 h

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.

Step 2 Graph the line.

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