# 7.6: Extracting the Equation from a Graph

**At Grade**Created by: CK-12

**Practice**Square and Cube Root Function Families

The graph of a cubic function starts at the point (2, 2). It passes through the point (10, -2). What is the equation of the function?

### Extracting the Equation from a Graph

In this concept, instead of graphing from the equation, we will now find the equation when we are given the graph.

Let's determine the equation of the graph below.

We know this is a square root function, so the general form is

The equation is

Now, let's find the equation of the cubed root function where

First, plug in what we know to the general equation;

The equation of the function is

Finally, let's find the equation of the function below.

It looks like

The equation of this function is

When finding the equation of a cubed root function, you may assume that one of the given points is

### Examples

#### Example 1

Earlier, you were asked to find the equation of the function.

First, plug in what we know to the general equation;

The equation of the function is

#### Example 2

Find the equation of the following function.

Substitute what you know into the general equation to solve for

The equation of this function is

#### Example 3

Find the equation of the following function.

Substitute what you know into the general equation to solve for

The equation of this function is

#### Example 4

Find the equation of a square root equation with a starting point of

Substitute what you know into the general equation to solve for \begin{align*}a\end{align*}. From the graph, the starting point, or \begin{align*}(h, k)\end{align*} is \begin{align*}(-5, -3)\end{align*} and \begin{align*}(4, -6)\end{align*} are a point on the graph.

\begin{align*}y&=a \sqrt{x+5}-3 \\ -6&=a \sqrt{4+5}-3 \\ -3&=3a \\ -1&=a\end{align*}

The equation of this function is \begin{align*}y=- \sqrt{x+5}-3\end{align*}.

### Review

Write the equation for each function graphed below.

- Write the equation of a square root function with starting point \begin{align*}(-6, -3)\end{align*} passing through \begin{align*}(10, -15)\end{align*}.
- Write the equation of a cube root function with \begin{align*}(h, k) = (2, 7)\end{align*} passing through \begin{align*}(10, 11)\end{align*}.
- Write the equation of a square root function with starting point \begin{align*}(-1, 6)\end{align*} passing through \begin{align*}(3, 16)\end{align*}.
- Write the equation of a cubed root function with \begin{align*}(h, k)=(-1, 6)\end{align*} passing through \begin{align*}(7, 16)\end{align*}.
- Write the equation of a cubed root function with \begin{align*}(h, k)=(7, 16)\end{align*} passing through \begin{align*}(-1, 6)\end{align*}.
- How do the two equations above differ? How are they the same?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.6.

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### Image Attributions

Here you'll look at the graph of a square root or cubed root function and determine the equation.

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