# Algebra II : Graphing Exponential Functions

## Example Questions

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### Example Question #1 : Graphing Exponential Functions

Give the -intercept of the graph of the equation .

The graph has no -intercept.

The graph has no -intercept.

Explanation:

Set  and solve for

We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of  has no -intercept.

### Example Question #2 : Graphing Exponential Functions

What is/are the asymptote(s) of the graph of the function ?

and

Explanation:

An exponential function of the form

has as its one and only asymptote the horizontal line

Since we define  as

,

then

and the only asymptote is the line of the equation .

### Example Question #3 : Graphing Exponential Functions

Determine whether each function represents exponential decay or growth.

a) growth

b) decay

a) decay

b) growth

a) growth

b) growth

a) decay

b) decay

a) decay

b) growth

Explanation:

a)

This is exponential decay since the base, , is between  and .

b)

This is exponential growth since the base, , is greater than .

### 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 #122 : Solving And Graphing Exponential Equations

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 #5 : 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 #762 : Exponents

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 #6 : 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 #7 : 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 #9 : 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,

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