SAT II Math I : Solving Exponential Functions

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

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Example Question #1 : Solving Exponential Functions

Solve the following function: 

Possible Answers:

 and 

Correct answer:

 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.  

This means your answer can be  or .

Example Question #1 : Asymptotes

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

Possible Answers:

Correct answer:

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

 ?

Possible Answers:

 

Correct answer:

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 #12 : Understanding Asymptotes

Find the vertical asymptote of the equation.

Possible Answers:

There are no vertical asymptotes.

Correct answer:

Explanation:

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

Example Question #1112 : Mathematical Relationships And Basic Graphs

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

Possible Answers:

One vertical asymptote at .

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

One horizontal asymptote at .

One vertical asymptote at .

Correct answer:

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.

Exp_asymp

Example Question #1123 : Mathematical Relationships And Basic Graphs

Solve the equation for .

Possible Answers:

Correct answer:

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 #47 : Exponents

Solve the equation for .

Possible Answers:

 

 

 

 

Correct answer:

 

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 #1 : 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.

Possible Answers:

Correct answer:

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 #2 : Solving Exponential Equations

Solve for .

Possible Answers:

Correct answer:

Explanation:

8 and 4 are both powers of 2.

Example Question #2 : Solving Exponential Functions

Solve for :

Possible Answers:

No solution

Correct answer:

Explanation:

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

Add 9 to both sides: 

Then, subtract 2x from both sides: 

Finally, divide both sides by 3: 

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