4.1 Linear functions  (Page 4/27)

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Are the units for slope always

Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.

Calculate slope

The slope, or rate of change, of a function $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ can be calculated according to the following:

where $\text{\hspace{0.17em}}{x}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{x}_{2}\text{\hspace{0.17em}}$ are input values, $\text{\hspace{0.17em}}{y}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}_{2}\text{\hspace{0.17em}}$ are output values.

Given two points from a linear function, calculate and interpret the slope.

1. Determine the units for output and input values.
2. Calculate the change of output values and change of input values.
3. Interpret the slope as the change in output values per unit of the input value.

Finding the slope of a linear function

If $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a linear function, and $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,1\right)\text{\hspace{0.17em}}$ are points on the line, find the slope. Is this function increasing or decreasing?

The coordinate pairs are $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,1\right).\text{\hspace{0.17em}}$ To find the rate of change, we divide the change in output by the change in input.

We could also write the slope as $\text{\hspace{0.17em}}m=0.6.\text{\hspace{0.17em}}$ The function is increasing because $\text{\hspace{0.17em}}m>0.$

If $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a linear function, and $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,4\right)\text{\hspace{0.17em}}$ are points on the line, find the slope. Is this function increasing or decreasing?

$m=\frac{4-3}{0-2}=\frac{1}{-2}=-\frac{1}{2};$ decreasing because $m<0.$

Finding the population change from a linear function

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The rate of change relates the change in population to the change in time. The population increased by $\text{\hspace{0.17em}}27,800-23,400=4400\text{\hspace{0.17em}}$ people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.

So the population increased by 1,100 people per year.

The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

$m=\frac{1,868-1,442}{2,012-2,009}=\frac{426}{3}=\text{142 people per year}$

Writing and interpreting an equation for a linear function

Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form    and the point-slope form    . Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] .

We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose $\text{\hspace{0.17em}}\left(0,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,4\right).\text{\hspace{0.17em}}$

$\begin{array}{ccc}\hfill m& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{4-7}{4-0}\hfill \\ & =& -\frac{3}{4}\hfill \end{array}$

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.

If we want to rewrite the equation in the slope-intercept form, we would find

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