2.1 Linear functions  (Page 6/17)

Writing and interpreting an equation for a linear function

Now that we have written equations for linear functions in both the slope-intercept form and the point-slope form, we can choose which method to use 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 $f$ 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 $\left(0,7\right)$ and $\left(4,4\right).$ We can use these points to calculate the slope.

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

If we wanted to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y -axis when the output value is 7. Therefore, $b=7.$ We now have the initial value $b$ and the slope $m$ so we can substitute $m$ and $b$ into the slope-intercept form of a line.

So the function is $f(x)=-\frac{3}{4}x+7,$ and the linear equation would be $y=-\frac{3}{4}x+7.$

Writing an equation for a linear function

Write an equation for a linear function given a graph of $f$ shown in [link] .

Identify two points on the line, such as $\left(0,2\right)$ and $\left(-2,\text{−4}\right).$ Use the points to calculate the slope.

$$\begin{array}{l}\begin{array}{l}\\ m=\frac{y_2-y_1}{x_2-x_1}\end{array}\hfill \\ =\frac{-4-2}{-2-0}\hfill \\ =\frac{-6}{-2}\hfill \\ =3\hfill \end{array}$$

Substitute the slope and the coordinates of one of the points into the point-slope form.

$$\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-\left(-4\right)=3\left(x-\left(-2\right)\right)\\ y+4=3\left(x+2\right)\end{array}$$

We can use algebra to rewrite the equation in the slope-intercept form.

$$\begin{array}{l}y+4=3(x+2)\hfill \\ y+4=3x+6\hfill \\ y=3x+2\hfill \end{array}$$

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