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# Write a Function in Slope-Intercept Form

## If y=mx+b and y=f(x) then use f(x)=mx+b to find coordinates on the line

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Write a Function in Slope-Intercept Form

What if the linear function W(g)\begin{align*}W(g)\end{align*} represented a family's monthly water bill, with g\begin{align*}g\end{align*} as the number of gallons of water used. If you know what the function was, could you find W(25)\begin{align*}W(25)\end{align*}? How about if you know the slope of the function and the value of W(25)\begin{align*}W(25)\end{align*}? Could you determine what the function is? Suppose you know the values of W(25)\begin{align*}W(25)\end{align*} and W(50)\begin{align*}W(50)\end{align*}. Could you determine the function in this case?

### Writing a Function in Slope-Intercept Form

Remember that a linear function has the form f(x)=mx+b\begin{align*}f(x)=mx+b\end{align*}. Here f(x)\begin{align*}f(x)\end{align*} represents the y\begin{align*}y\end{align*} values of the equation or the graph. So y=f(x)\begin{align*}y=f(x)\end{align*} and they are often used interchangeably. Using the functional notation in an equation often provides you with more information.

For instance, the expression f(x)=mx+b\begin{align*}f(x)=mx+b\end{align*} shows clearly that x\begin{align*}x\end{align*} is the independent variable because you substitute values of x\begin{align*}x\end{align*} into the function and perform a series of operations on the value of x\begin{align*}x\end{align*} in order to calculate the values of the dependent variable, y\begin{align*}y\end{align*}.

In this case when you substitute x\begin{align*}x\end{align*} into the function, the function tells you to multiply it by m\begin{align*}m\end{align*} and then add b\begin{align*}b\end{align*} to the result. This process generates all the values of y\begin{align*}y\end{align*} you need.

Let's consider the function f(x)=3x4\begin{align*}f(x)=3x-4\end{align*} and find f(2),f(0),\begin{align*}f(2), f(0),\end{align*} and f(1)\begin{align*}f(-1)\end{align*}:

Each number in parentheses is a value of x\begin{align*}x\end{align*} that you need to substitute into the equation of the function.

f(2)=2;f(0)=4; and f(1)=7\begin{align*}f(2)=2; f(0)=-4; \ and \ f(-1)=-7\end{align*}

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, f(1)=7\begin{align*}f(-1)=-7\end{align*}. This means the ordered pair (–1, –7) is a solution to f(x)=3x4\begin{align*}f(x)=3x-4\end{align*} and appears on the graphed line. You can use this information to write an equation for a function.

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

1. A line with m=3.5\begin{align*}m=3.5\end{align*} and f(2)=1\begin{align*}f(-2)=1\end{align*}

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.

Substitute the value for the slope.Use the ordered pair to solve for b.Rewrite the equation.oryy1byf(x)=mx+b=3.5x+b=3.5(2)+b=8=3.5x+8=3.5x+8\begin{align*}&& y& =mx+b\\ \text{Substitute the value for the slope.} && y& =3.5x+b\\ \text{Use the ordered pair to solve for} \ b. && 1& =3.5(-2)+b\\ && b& =8\\ \text{Rewrite the equation.} && y& =3.5x+8 \\ \text{or} && f(x)& =3.5x+8\end{align*}

1. A line with f(1)=2\begin{align*}f(-1)=2\end{align*} and f(5)=20\begin{align*}f(5)=20\end{align*}

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

First, you must find the slope:

m=y2y1x2x1=2025(1)=186=3\begin{align*}m=\frac{y_2-y_1}{x_2-x_1}=\frac{20-2}{5-(-1)}=\frac{18}{6}=3\end{align*}.

Now use the slope-intercept form:

Substitute the value for the slope.Use the ordered pair to solve for b.Rewrite the equation.oryy2byf(x)=mx+b=3x+b=3(1)+b=5=3x+5=3x+5\begin{align*}&& y& =mx+b\\ \text{Substitute the value for the slope.} && y& =3x+b\\ \text{Use the ordered pair to solve for} \ b. && 2& =3(-1)+b\\ && b& =5\\ \text{Rewrite the equation.} && y& =3x+5\\ \text{or} && f(x)& =3x+5\end{align*}

### Examples

#### Example 1

Earlier, you were told that the linear function W(g)\begin{align*}W(g)\end{align*} represented a family's monthly water bill, with g\begin{align*}g\end{align*} as the number of gallons of water used. If you know what the function is, could you find W(25)\begin{align*}W(25)\end{align*}? If you know the slope of the function and the value of W(25)\begin{align*}W(25)\end{align*}, could you determine what the function is? If you know the values of W(25)\begin{align*}W(25)\end{align*} and W(50)\begin{align*}W(50)\end{align*}, could you determine what the function is?

If you were given the function, to find W(25)\begin{align*}W(25)\end{align*}, you would just plug in 25 for g\begin{align*}g\end{align*}, the independent value and then simplify. If you knew the slope and the value of W(25)\begin{align*}W(25)\end{align*}, you could find the function by treating 25 and the value of W(25)\begin{align*}W(25)\end{align*} as a point. If you knew the values of W(25)\begin{align*}W(25)\end{align*} and W(50)\begin{align*}W(50)\end{align*}, you know two points on the line and could find the function.

#### Example 2

Write an equation for a line with f(0)=2\begin{align*}f(0)=2\end{align*} and f(3)=4\begin{align*}f(3)=-4\end{align*} and use it to find f(5),f(2),f(0), and f(z)\begin{align*}f(-5), f(2), f(0), \text{ and } f(z)\end{align*}.

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 y\begin{align*}y\end{align*}-intercept. So, all we have to do is find the slope and then plug both values into the slope-intercept form:

m=y2y1x2x1=4230=63=2\begin{align*}m=\frac{y_2-y_1}{x_2-x_1}=\frac{-4-2}{3-0}=\frac{-6}{3}=-2\end{align*}.

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

1. 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*}.
2. 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.

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

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

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