Find the Function of a Graph

It’s important, when working with linear functions, to be able to find the equation of a linear function by reading a graph of the function.

It’s actually a lot easier than it sounds, as long as you remember that all linear functions graph as straight lines that are written as $f (x) = a x + b$ , where $a$ is the slope and $b$ $y$ -intercept (the constant term). Here are instructions for how to do it:

Rule

Finding the Equation of a Function from a Graph

1.: Calculate the slope $a$ , either by the formula for the slope, or by manually counting how much $y$ increases or decreases by when you move $x$ one place to the right.
2.: Find the $y - i n t e r c e p t$ $b$ by locating where the graph intersects the $y$ -axis.
3.: Insert the values into the expression $f (x) = a x + b$ .

Example 1

Find the equation of the function from the graph.

1.: Use the formula for the slope: $\begin{array}{l} a = \frac{0 - 16}{4 - 0} = \frac{- 16}{4} = - 4 \end{array}$
You can also find the answer graphically by moving along the $x$ -axis one place to the right. You can see that the $y$ -value goes from 16 to 12. That means it decreases by $16 - 12 = 4$ . Thus, the slope is $- 4$ .
2.: Find the $y$ -intercept, $b$ , on the graph. From the drawing, you can see that the graph intersects the $y$ -axis at $y = 16$ .
3.: Insert the values into the expression $f (x) = a x + b$ .
4.: You get $f (x) = - 4 x + 16$ , the function of this graph.

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Activities 2

Activities 3