# SAT II Math I : Solving Exponential Functions

## Example Questions

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### Example Question #1 : Solving Exponential Functions

Solve the following function:

and

and

Explanation:

You must get  by itself so you must add  to both side which results in

.

You must get the square root of both side to undue the exponent.

This leaves you with .

But since you square the  in the equation, the original value you plug can also be its negative value since squaring it will make it positive anyway.

### 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 : Solving And Graphing Exponential Equations

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 #4 : Asymptotes

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

One horizontal asymptote at .

No asymptotes.  goes to positive  infinity in both the  and  directions.

One vertical asymptote at .

One vertical asymptote at .

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 Functions

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

.

### Example Question #4 : Solving Exponential Equations

Solve for .

Explanation:

8 and 4 are both powers of 2.

### Example Question #5 : Solving Exponential Equations

Solve for :

No solution

Explanation:

Because both sides of the equation have the same base, set the terms equal to each other.