Precalculus : Exponential Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Solve Exponential Equations

Solve:   

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

Example Question #2 : Solve Exponential Equations

Solve the equation for .

Possible Answers:

Correct answer:

Explanation:

The key to this is that . From here, the equation can be factored as if it were .

 and 

 and 

 and 

Now take the natural log (ln) of the two equations.

 and 

 and 

 

Example Question #1 : Graph Exponential Functions

Choose the description below that matches the equation: 

Possible Answers:

Exponential growth 

Y-intercept at 

Exponential growth 

Y-intercept at 

Exponential growth 

Y-intercept at 

Exponential decay 

 

Y-intercept at 

Exponential decay 

Y-intercept at 

Correct answer:

Exponential growth 

Y-intercept at 

Explanation:

Exponential graphs can either decay or grow. This is based on the value of the base of the exponent. If the base is greater than , the graph will be growth. And, if the base is less than , then the graph will be decay. In this situation, our base is . Since this is greater than , we have a growth graph. Then, to determine the y-intercept we substitute . Thus, we get: 

 for the y-intercept. 

Example Question #2 : Graph Exponential Functions

Choose the description that matches the equation below: 

Possible Answers:

Exponential growth 

Exponential decay 

Exponential decay 

Exponential growth 

Exponential decay 

Correct answer:

Exponential decay 

Explanation:

Exponential graphs can either decay or grow. This is based on the value of the base of the exponent. If the base is greater than , the graph will be growth. And, if the base is less than , then the graph will be decay. In this situation, our base is . Since this is less than , we have a decay graph. Then, to determine the y-intercept we substitute . Thus, we get: 

 for the y-intercept. 

Example Question #3 : Graph Exponential Functions

Which of the following represents the graph of ?

Possible Answers:



Graph4


Graph2

Graph3

Graph1

Correct answer:

Graph1

Explanation:

Note that the negative sign in this function comes outside of the parentheses. This should show you that the bigger the number in parentheses, the lower the curve of the graph will go.  Since this is an exponential function, the larger that the x value gets, then, the "more negative" this graph will go. The graph closest to zero on the left-hand side - where x is negative - and then shoots down and to the right rapidly when x gets larger is the correct graph.

Example Question #4 : Graph Exponential Functions

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

Correct answer:

Explanation:

The -coordinate ofthe -intercept of the graph of  is 0, and its -coordinate is :

The -intercept is the point .

Example Question #21 : Solving Exponential Functions

Does the function  have any -intercepts? 

Possible Answers:

Yes,  and 

No 

Yes, 

That cannot be determined from the information given. 

Yes, 

Correct answer:

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 #10 : Graphing Exponential Functions

Which of the following functions represents exponential decay? 

Possible Answers:

Correct answer:

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 #8 : Graphing Exponential Functions

What is the -intercept of ?

Possible Answers:

There is no -intercept. 

Correct answer:

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: 

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