7.2 Linear Functions And Their Graphs

Objectives

1. Use intercepts to graph a linear equation.

2. Calculate slope.3. Use the slope and y-intercept to

graph a line.4. Graph horizontal and vertical

lines.5. Interpret slope as a rate of

change.6. 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 = 4x-intercept—point where the

line crosses the x-axisY-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 EquationGraph: 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.

12

12

run

rise

in Change

in Change

xx

yy

x

y

ExampleFind 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.

2 1

2 1

Change in 4 ( 1) 55.

Change in 2 ( 3) 1

y yy

mx x x

The Slope-intercept form of the Equation of a Line

Recall: Ax + By = C is an equation of a line.Solving for y, we getBy = -Ax + Cy = (-Ax + C)/By = (-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 -Am = ---------- = ------- = --- · ----- = -A/B 0 – C/A -C/A B C

(0, C/B)

(C/A, 0)

The Slope-intercept form of the Equation of a Liney = (-A/B)x + C/Bm = -A/B

y = mx + bwhere m = slope b = y-intercept

E.g., y = (3/2)x + 4 m = 3/2 b = 4

(0, C/B)

(C/A, 0)

ExampleGraph: 2x + 5y = 0 using the

slope and y-intercept.Solution:

2x + 5y = 05y = -2xy = (-2/5)x

m = -2/5b = 0

Your TurnGraph the lines using the slope-

intercept form of the equation.1. y = (-3/4)x – 52. 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 LineSlope: ration of a change in y to a

corresponding change in xm = 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.

ExampleSolution:

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.