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 starting point is
. Plugging this in for
and
, we have
. Now, find
, using the given point,
. Let’s substitute it in for
and
and solve for
.

The equation is .

#### Example B

Find the equation of the cubed root function where and and passes through .

**
Solution:
**
First, plug in what we know to the general equation;
. Now, substitute
and
and solve for
.

The equation of the function is .

#### Example C

Find the equation of the function below.

**
Solution:
**
It looks like
is
. Plug this in for
and
and then use the second point to find
.

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 . Whichever point is on the “bend” is for the purposes of this text.

**
Intro Problem Revisit
**

First, plug in what we know to the general equation; . Now, substitute and and solve for .

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 and passes through .

#### Answers

1. Substitute what you know into the general equation to solve for . From Example C, you may assume that is and is .

The equation of this function is .

2. Substitute what you know into the general equation to solve for . From the graph, the starting point, or is and are a point on the graph.

The equation of this function is .

3. Substitute what you know into the general equation to solve for . From the graph, the starting point, or is and are a point on the graph.

The equation of this function is .

### Explore More

Write the equation for each function graphed below.

- Write the equation of a square root function with starting point passing through .
- Write the equation of a cube root function with passing through .
- Write the equation of a square root function with starting point passing through .
- Write the equation of a cubed root function with passing through .
- Write the equation of a cubed root function with passing through .
- How do the two equations above differ? How are they the same?