Algebra: Graphs, Functions, and Linear Systems CHAPTER 7 Algebra: Graphs, Functions, and Linear Systems

Linear Functions and Their Graphs 7.2 Linear Functions and Their Graphs

Objectives Use intercepts to graph a linear equation. Calculate slope. Use the slope and y-intercept to graph a line. Graph horizontal and vertical lines. Interpret slope as a rate of change. Use slope and y-intercept to model data.

Graphing Using Intercepts All equations of the form Ax + By = C are straight lines when graphed, as long as A and B are not both zero, and are called linear equations in two variables.

Example: Using Intercepts to Graph a Linear Equation Graph: 3x + 2y = 6. Solution: Find the x-intercept by letting y = 0 and solving for x. 3x + 2y = 6 3x + 2 · 0 = 6 3x = 6 x = 2 Find the y-intercept by letting x = 0 and solving for y. 3x + 2y = 6 3 · 0 + 2y = 6 2y = 6 y = 3

Example continued The x-intercept is 2, so the line passes through the point (2,0). The y-intercept is 3, so the line passes through the point (0,3). Now, we verify our work by checking for x = 1. Plug x = 1 into the given linear equation. We leave this to the student. For x = 1, the y-coordinate should be 1.5.

Slope The slope of the line through the distinct points (x1,y1) and (x2,y2) is where x2 – x1 ≠ 0.

Example: Using the Definition of Slope Find the slope of the line passing through the pair of points: (−3, −1) and (−2, 4). Solution: Let (x1, y1) = (−3, −1) and (x2, y2) = (−2, 4). We obtain the slope such that Thus, the slope of the line is 5.

The Slope-Intercept Form of the Equation of a Line The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is y = mx + b.

The Slope-Intercept Form of the Equation of a Line Graphing y = mx + b using the slope and y-intercept: Plot the point containing the y-intercept on the y-axis. This is the point (0, b). Obtain a second point using the slope m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point. Use a straightedge to draw a line through the two points. Draw arrowheads at the end of the line to show that the line continues indefinitely in both directions.

Example: Graphing by Using the Slope and y-intercept Graph the linear function by using the slope and y-intercept. Solution: Since the graph is given in slope-intercept form we can easily find the slope and y-intercept.

Example continued Step 1 Plot the point containing the y-intercept on the y-axis. The y-intercept is (0, 2). Step 2 Obtain a second point using the slope, m. The slope as a fraction is already given: We plot the second point at (3, 4). Step 3 Use a straightedge to draw a line through the two points.

Example: Graphing by Using the Slope and the y-intercept Graph the linear function 2x + 5y = 0 by using the slope and y-intercept. Solution: We put the equation in slope-intercept form by solving for y. slope-intercept form

Example continued Next, we find the slope and y-intercept: Start at y-intercept (0, 0) and obtain a second point by using the slope. We obtain (5, −2) as the second point and use a straightedge to draw the line through these points.

Equations of Horizontal and Vertical Lines The graph of y = b or f(x) = b is a horizontal line. The y-intercept is b. The graph of x = a is a vertical line. The x-intercept is a.

Example: Graphing a Horizontal Line Graph y = −4 in the rectangular coordinate system. Solution: All ordered pairs have y-coordinates that are −4. Any value can be used for x. We graph the three ordered pairs in the table: (−2,−4), (0, −4), and (3,−4). Then use a straightedge to draw the horizontal line.

Example continued The graph of y = −4 or f(x) = −4.

Example: Graphing a Vertical Line Graph x = 2 in the rectangular coordinate system. Solution: All ordered pairs have the x-coordinate 2. Any value can be used for y. We graph the ordered pairs (2,−2), (2,0), and (2,3). Drawing a line that passes through the three points gives the vertical line.

Example continued The graph of x = 2. No vertical line represents a linear function. All other lines are graphs of functions.

Horizontal and Vertical Lines

Slope as Rate of Change Slope is defined as a ratio of a change in y to a corresponding change in x. Slope can be interpreted as a rate of change in an applied situation.

Example: Slope as a Rate of Change The line graphs in figure show the percentage of American men and women ages 20 to 24 who were married from 1970 through 2010. Find the slope of the line segment representing women. Describe what the slope represents.

Example continued Solution: We let x represent a year and y the percentage of married women ages 20–24 in that year. The two points shown on the line segment for women have the following coordinates (1970, 65) and (2010, 21). The slope indicates that for the period from 1970 through 2010, the percentage of married women ages 20 to 24 decreased by 1.1 per year. The rate of change is –1.1% per year.

Modeling Data with the Slope-Intercept Form of the Equation of a Line Linear functions are useful for modeling data that fall on or near a line.