# Algebra II : Solving and Graphing Exponential Equations

## Example Questions

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### Example Question #4 : Graphing Exponential Functions

Match each function with its graph.

1.

2.

3.

a.

b.

c.

1.

2.

3.

1.

2.

3.

1.

2.

3.

1.

2.

3.

1.

2.

3.

Explanation:

For , our base is greater than  so we have exponential growth, meaning the function is increasing. Also, when , we know that  since . The only graph that fits these conditions is .

For , we have exponential growth again but when . This is shown on graph .

For , we have exponential decay so the graph must be decreasing. Also, when . This is shown on graph .

### Example Question #5 : Graphing Exponential Functions

An exponential funtion  is graphed on the figure below to model some data that shows exponential decay. At  is at half of its initial value (value when ). Find the exponential equation of the form  that fits the data in the graph, i.e. find the constants  and .

Explanation:

To determine the constant , we look at the graph to find the initial value of  , (when ) and find it to be .  We can then plug this into our equation  and we get . Since , we find that .

To find , we use the fact that when  is one half of the initial value . Plugging this into our equation with  now known gives us  . To solve for , we make use the fact that the natural log is the inverse function of , so that

.

We can write our equation as   and take the natural log of both sides to get:

or .

Then .

Our model equation is .

### Example Question #1 : Graphing Exponential Functions

In 2010, the population of trout in a lake was 416. It has increased to 521 in 2015.

Write an exponential function of the form  that could be used to model the fish population of the lake. Write the function in terms of , the number of years since 2010.

Explanation:

We need to determine the constants  and . Since  in 2010 (when ), then  and

To get , we find that when .  Then  .

Using a calculator, , so .

Then our model equation for the fish population is

### Example Question #7 : Graphing Exponential Functions

What is the -intercept of the graph ?

Explanation:

The -intercept of any graph describes the -value of the point on the graph with a -value of .

Thus, to find the -intercept substitute .

In this case, you will get,

### Example Question #8 : Graphing Exponential Functions

What is the -intercept of ?

There is no -intercept.

Explanation:

The -intercept of a graph is the point on the graph where the -value is .

Thus, to find the -intercept, substitute  and solve for .

Thus, we get:

### Example Question #9 : Graphing Exponential Functions

What is the -intercept of

Explanation:

The -intercept of any function describes the point where .

Substituting this in to our funciton, we get:

### Example Question #10 : Graphing Exponential Functions

Which of the following functions represents exponential decay?

Explanation:

Exponential decay describes a function that decreases by a factor every time  increases by .

These can be recognizable by those functions with a base which is between  and .

The general equation for exponential decay is,

where the base is represented by  and .

Thus, we are looking for a fractional base.

The only function that has a fractional base is,

### Example Question #21 : Solving Exponential Functions

Does the function  have any -intercepts?

Yes,  and

No

Yes,

That cannot be determined from the information given.

Yes,

No

Explanation:

The -intercept of a function is where . Thus, we are looking for the -value which makes .

If we try to solve this equation for  we get an error.

To bring the exponent down we will need to take the natural log of both sides.

Since the natural log of zero does not exist, there is no exponent which makes this equation true.

Thus, there is no -intercept for this function.

### Example Question #31 : Solving Exponential Functions

Which of the following correctly describes the graph of an exponential function with a base of three?

It begins by decreasing gradually and then decreases more quickly.

It begins by decreasing quickly and then levels out.

It starts out by gradually increasing and then increases faster and faster.

It starts by increasing quickly and then levels out.

It stays constant.