Linear Functions and Modeling

Graphing Linear Functions

Linear Functions and Slope

A linear function is written in the form f(x)=mx+bf(x) = mx + b, where mm is the slope and bb is the yy-intercept of the function. The slope is also called the rate of change.
A linear function is a function whose graph is a line. All linear functions can be written in the form:
f(x)=mx+bf(x) = mx + b
The variable mm is the slope. The variable bb is the yy-intercept of the function. The slope is the ratio of the change in yy to the change in xx for two points on a line, represented as:
RiseRun\frac{\text{Rise}}{\text{Run}}
  • The rise is the vertical change between two points on a line.
  • The run is the horizontal change between two points on a line.
The y-intercept is the value of yy where a graph touches or crosses the yy-axis. Linear functions can also be written in slope-intercept form using yy in place of f(x)f(x):
y=mx+by=mx+b
Step-By-Step Example
Graphing a Linear Function
Graph the linear function:
f(x)=2x+3f(x)=2x+3
Step 1

Determine the slope and intercept of the line.

The slope is the coefficient of xx, or m=2m=2.

The yy-intercept is the constant term, which is also the value of f(0)f(0), or b=3b=3.
Step 2
Plot the yy-intercept at (0,3)(0, 3), and then use the slope to locate another point.
Slope=RiseRunSlope=21\begin{aligned}\text{Slope}&=\frac{\text{Rise}}{\text{Run}}\\ \text{Slope}&=\frac{2}{1}\end{aligned}
Move up 2 units and right 1 unit. Connect the points with a line.
Solution
Graph the linear function as shown.
Step-By-Step Example
Determining the Slope and yy-Intercept of a Line
Identify the slope and yy-intercept of:
y+5x=8y + 5x =8
Step 1
The slope-intercept form of a line, where mm is the slope and bb is the yy-intercept is:
y=mx+by=mx+b
Rewrite the given equation in slope-intercept form by subtracting 5x5x from both sides.
y+5x=8y5x=5x+8y=5x+8\begin{aligned}y+5x&=\phantom{-}8\\\underline{\phantom{y}- 5x}&\phantom{=}\underline{-5x\phantom{+8}}\\y&=-5x+8\end{aligned}
Step 2

Identify the slope.

The slope is the coefficient of the xx-term.
y=5x+8y={\color{#c42126}{-5}}x+8
The slope is –5.
Step 3

Identify the yy-intercept.

The yy-intercept is the constant term.
y=5x+8y=-5x+{\color{#c42126}{8}}
The yy-intercept is 8.
Solution
The slope, or mm, is –5. The yy-intercept, or bb, is 8.

Rate of Change of a Linear Function

A linear function has a constant rate of change equal to the slope of its graph.
Rate of change describes how one value changes in relation to how another value changes. Since slope is rise over run, or the change in yy-values over the change in xx-values, the slope of a linear function can be interpreted as the rate of change. In real-world situations, this may represent a rate such as speed or unit cost.
Step-By-Step Example
Comparing Rates of Change

Suppose two cars are traveling at constant speeds. Car 1 travels at 90 kilometers per hour (km/h), while Car 2 travels at 100 km/h.

Graph the distance traveled over time for the two cars. Then compare the rates of change.

Step 1
Write a linear function for each car that represents the distance traveled. Use the formula:
Distance = (Rate) (Time)\text{Distance = (Rate) (Time)}
Let distance be f(x)f(x). Rate is the speed that each car travels. Let time be xx.
  • Car 1: f(x)=90xf(x)=90x
  • Car 2: f(x)=100xf(x)=100x
Step 2
The equations for the functions are in slope-intercept form, where mm is the slope and bb is the yy-intercept:
y=mx+by = mx + b
The line for Car 1 has a slope of 90.
f(x)=90xf(x)={\color{#c42126}{90}}x
The line for Car 2 has a slope of 100.
f(x)=100xf(x)={\color{#c42126}{100}}x
Both functions have a yy-intercept of zero.
Step 3

Determine two points on each linear function by using the slope and yy -intercept.

For Car 1:

The yy -intercept is zero. So, one of the points is (0,0)(0, 0).

From (0,0)(0, 0), use the slope of 90 to find another point. The rise is 90, and the run is 1. So, the second point is at (1,90)(1, 90).

For Car 2:

The yy-intercept is zero. So, one of the points is (0,0)(0, 0).

From (0,0)(0, 0), use the slope of 100 to find another point. The rise is 100, and the run is 1. So the other point is (1,100)(1, 100).

Step 4
Graph the functions.
Solution
The lines on the graph represent the distance that each car travels. The line for Car 2 is steeper than the line for Car 1. So, the rate of change for Car 2 is greater than the rate of change for Car 1.