# 7.6: Extracting the Equation from a Graph

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

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?

### Guidance

This concept is the opposite of the previous two. Instead of graphing from the equation, we will now find the equation, given the graph.

#### Example A

Determine the equation of the graph below.

**Solution:** From the previous two concepts, we know this is a square root function, so the general form is

The equation is

#### Example B

Find the equation of the cubed root function where

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

The equation of the function is

#### Example C

Find the equation of the function below.

**Solution:** 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

**Intro Problem Revisit**

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

The equation of the function is

### Guided Practice

Find the equation of the functions below.

1.

2.

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

#### Answers

1. Substitute what you know into the general equation to solve for

The equation of this function is

2. Substitute what you know into the general equation to solve for

The equation of this function is \begin{align*}y=4 \sqrt{x-4}-11\end{align*}.

3. 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*}.

### Explore More

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?

### Image Attributions

## Description

## Learning Objectives

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

## Difficulty Level:

At Grade## Subjects:

## Concept Nodes:

## Date Created:

Mar 12, 2013## Last Modified:

Jun 04, 2015## Vocabulary

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