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.

Using Intercepts to Graph Lines Equation Equation of a Line: Ax + By = C E.g., 2x + 3y = 4 x-intercept—point where the line crosses the x-axis Y-intercept—point where the line crosses the y-axis To find x-intercept, let y = 0 and solve for x. To find y-intercept, let x = 0 and solve for y. (0, y0) (x0, 0)

Example: Using Intercepts to Graph Lines 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: Using Intercepts to Graph Lines Equation The x-intercept is 2; line passes through (2,0). The y-intercept is 3; line passes through (0,3). Now, we verify our work by checking for x = 1. Plug x = 1 into the given linear equation. For x = 1, the y-coordinate should be 1.5.

Slope of a Line Slope of the line through points (x1,y1) and (x2,y2): where x2 – x1 ≠ 0.

Example Find the slope of the line passing through points: (−3, −1) and (−2, 4). Solution: Let (x1, y1) = (−3, −1) and (x2, y2) = (−2, 4). Thus, the slope of the line is 5.

The Slope-intercept form of the Equation of a Line Recall: Ax + By = C is an equation of a line. Solving for y, we get By = -Ax + C y = (-Ax + C)/B y = (-A/B)x + C/B When x = 0, y = C/B y = 0, x = (C/B)/(A/B) = C/A C/B – 0 C/B C -A m = ---------- = ------- = --- · ----- = -A/B 0 – C/A -C/A B C (0, C/B) (C/A, 0)

The Slope-intercept form of the Equation of a Line y = (-A/B)x + C/B m = -A/B y = mx + b where m = slope b = y-intercept E.g., y = (3/2)x + 4 m = 3/2 b = 4 (0, C/B) (C/A, 0)

Example Graph: 2x + 5y = 0 using the slope and y- intercept. Solution: 2x + 5y = 0 5y = -2x y = (-2/5)x m = -2/5 b = 0

Your Turn Graph the lines using the slope-intercept form of the equation. y = (-3/4)x – 5 2x + 3y + 4 = 0

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

Interpretation of the Slope of a Line Slope: ration of a change in y to a corresponding change in x m = Δx/Δy Slope can be interpreted as a rate of change in the vertical value as the corresponding horizontal value changes.

Interpretation of the Slope of a Line The graph shows cost of entitlement programs, in billions of dollars, from 2007 with projections through 2016. Find the slope of the line segment representing Social Security. Round to one decimal place. Describe what the slope represents.

Example Solution: Let x represent a year and y the cost (in $109).

Example (cont.) The slope indicates that for the period from 2007 through 2016, the cost of Social Security is projected to increase by approximately $43.1 billion per year. The rate of change is approximately $43.1 billion per year.