# Algebra II : Solving and Graphing Exponential Equations

## Example Questions

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### Example Question #1 : Asymptotes

What is the horizontal asymptote of the graph of the equation  ?

Explanation:

The asymptote of this equation can be found by observing that  regardless of . We are thus solving for the value of as approaches zero.

So the value that  cannot exceed is , and the line  is the asymptote.

### Example Question #2 : Asymptotes

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

?

Explanation:

An exponential equation of the form  has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

### Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Find the vertical asymptote of the equation.

There are no vertical asymptotes.

Explanation:

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

### Example Question #3 : Asymptotes

Determine the asymptotes, if any:

Explanation:

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the  term canceled, the  term will have a hole instead of an asymptote.

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:

### Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Which of the choices represents asymptote(s), if any?

Explanation:

Factor the numerator and denominator.

Notice that the  terms will cancel.  The hole will be located at  because this is a removable discontinuity.

The denominator cannot be equal to zero.  Set the denominator to find the location where the x-variable cannot exist.

The asymptote is located at .

### Example Question #4 : Asymptotes

Where is an asymptote located, if any?

Explanation:

Factor the numerator and denominator.

Rewrite the equation.

Notice that the  will cancel.  This means that the root of  will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

An asymptote is located at:

### Example Question #1 : Solving Exponential Functions

Consider the exponential function . Determine if there are any asymptotes and where they lie on the graph.

There is one horizontal asymptote at .

There is one vertical asymptote at .

There are no asymptotes.  goes to positive infinity in both the  and  directions.

There is one vertical asymptote at .

There is one horizontal asymptote at .

Explanation:

For positive  values,  increases exponentially in the  direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative  values, as  decreases, the term  becomes closer and closer to zero so  approaches  as we move along the negative  axis. As the graph below shows, this is forms a horizontal asymptote.

### Example Question #1 : Solving Exponential Equations

Solve the equation for .

Explanation:

Begin by recognizing that both sides of the equation have a root term of .

Using the power rule, we can set the exponents equal to each other.

### Example Question #1 : Solving Exponential Equations

Solve the equation for .

Explanation:

Begin by recognizing that both sides of the equation have the same root term, .

We can use the power rule to combine exponents.

Set the exponents equal to each other.

### Example Question #3 : Solving Exponential Equations

In 2009, the population of fish in a pond was 1,034. In 2013, it was 1,711.

Write an exponential growth function of the form  that could be used to model , the population of fish, in terms of , the number of years since 2009.

Explanation:

Solve for the values of and b:

In 2009,  and  (zero years since 2009). Plug this into the exponential equation form:

. Solve for  to get  .

In 2013,  and . Therefore,

or  .   Solve for  to get

.

Then the exponential growth function is

.

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