What if the linear function

### Writing a Function in Slope-Intercept Form

Remember that a linear function has the form

For instance, the expression **substitute** values of

In this case when you substitute

Let's consider the function

Each number in parentheses is a value of

Function notation tells you much more than the value of the independent variable. It also indicates a point on the graph. For example, in the above example,

#### Let's write an equation in slope-intercept form for each of the following lines:

- A line with
m=3.5 andf(−2)=1

You know the slope, and you know a point on the graph, (–2, 1). Using the methods presented in this Concept, write the equation for the line.

Begin with slope-intercept form.

- A line with
f(−1)=2 andf(5)=20

You know two points on the graph. Find the slope, and write the equation for the line.

First, you must find the slope:

Now use the slope-intercept form:

### Examples

#### Example 1

Earlier, you were told that the linear function

If you were given the function, to find

#### Example 2

Write an equation for a line with

Notice that the first point given as an input value is 0, and the output is 2, which means the point is (0,2). This is the

Now use the slope-intercept form.

\begin{align*}&& y& =mx+b\\ \text{Substitute the value for the slope.} && y& =-2x+b\\ \text{Substitute the value for the y-intercept} && y& =-2x+2\\ \text{or} && f(x)& =-2x+2\end{align*}

Now we find the values of \begin{align*}f(-5), f(2), f(0), \text{ and } f(z)\end{align*} for \begin{align*} f(x) =-2x+2 \end{align*}.

\begin{align*} f(-5) =-2(-5)+2=10+2=12 \end{align*}

\begin{align*} f(2) =-2(2)+2=-2 \end{align*}

\begin{align*} f(0) =-2(0)+2=0 \end{align*}

\begin{align*} f(z) =-2z+2 \end{align*}

### Review

- Consider the function \begin{align*}f(x)=-2x-3.\end{align*}
*Find*\begin{align*}f(-3), f(0),\end{align*}*and*\begin{align*}f(5)\end{align*}. - Consider the function \begin{align*}f(x)=\frac{2}{3}x+10.\end{align*}
*Find*\begin{align*}f(-9), f(0),\end{align*}*and*\begin{align*}f(9)\end{align*}.

In 3–10, find the equation of the linear function in slope–intercept form.

- \begin{align*}m=5, f(0)=-3\end{align*}
- \begin{align*}m=-2\end{align*}, \begin{align*}f(0)=5\end{align*}
- \begin{align*}m=-7, f(2)=-1\end{align*}
- \begin{align*}m=\frac{1}{3}, f(-1)=\frac{2}{3}\end{align*}
- \begin{align*}m=4.2, f(-3)=7.1\end{align*}
- \begin{align*}f\left (\frac{1}{4}\right )=\frac{3}{4}, f(0)=\frac{5}{4}\end{align*}
- \begin{align*}f(1.5)=-3, f(-1)=2\end{align*}
- \begin{align*}f(-1)=1\end{align*}, \begin{align*}f(1)=-1\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.3.