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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) represented a family's monthly water bill, with g as the number of gallons of water used. If you know what the function was, could you find W(25)? How about if you know the slope of the function and the value of W(25)? Could you determine what the function is? Suppose you know the values of W(25) and W(50). 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. Here f(x) represents the y values of the equation or the graph. So y=f(x) 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 shows clearly that x is the independent variable because you substitute values of x into the function and perform a series of operations on the value of x in order to calculate the values of the dependent variable, y.

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

Let's consider the function f(x)=3x4 and find f(2),f(0), and f(1):

Each number in parentheses is a value of x that you need to substitute into the equation of the function.

f(2)=2;f(0)=4; and f(1)=7

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. This means the ordered pair (–1, –7) is a solution to f(x)=3x4 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 and f(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.

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

  1. A line with f(1)=2 and f(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:

m=y2y1x2x1=2025(1)=186=3.

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

Examples

Example 1

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

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

Example 2

Write an equation for a line with f(0)=2 and f(3)=4 and use it to find f(5),f(2),f(0), and f(z).

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-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.

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*}

Review (Answers)

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

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Vocabulary

TermDefinition
undefined slope An undefined slope cannot be computed. Vertical lines have undefined slopes.

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