Advanced Geometry : How to graph an exponential function

Study concepts, example questions & explanations for Advanced Geometry

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Example Questions

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Example Question #1 : How To Graph An Exponential Function

Give the -intercept(s) of the graph of the equation 

Possible Answers:

The graph has no -intercept.

Correct answer:

Explanation:

Set  and solve for :

Example Question #2 : Graphing An Exponential Function

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

The graph of  has no -intercept. 

Correct answer:

The graph of  has no -intercept. 

Explanation:

Since the -intercept is the point at which the graph of  intersects the -axis, the -coordinate is 0, and the -coordinate can be found by setting  equal to 0 and solving for . Therefore, we need to find  such that . However, any power of a positive number must be positive, so  for all real , and  has no real solution. The graph of  therefore has no -intercept.

Example Question #2 : How To Graph An Exponential Function

Define a function  as follows:

Give the vertical aysmptote of the graph of .

Possible Answers:

The graph of  does not have a vertical asymptote.

Correct answer:

The graph of  does not have a vertical asymptote.

Explanation:

Since any number, positive or negative, can appear as an exponent, the domain of the function  is the set of all real numbers; in other words,  is defined for all real values of . It is therefore impossible for the graph to have a vertical asymptote.

Example Question #4 : Graphing An Exponential Function

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

The graph of  has no -intercept. 

Correct answer:

Explanation:

Since the -intercept is the point at which the graph of  intersects the -axis, the -coordinate is 0, and the -coordinate can be found by setting  equal to 0 and solving for . Therefore, we need to find  such that 

The -intercept is therefore .

Example Question #4 : How To Graph An Exponential Function

Define a function  as follows:

Give the horizontal aysmptote of the graph of .

Possible Answers:

Correct answer:

Explanation:

The horizontal asymptote of an exponential function can be found by noting that a positive number raised to any power must be positive. Therefore,  and  for all real values of . The graph will never crosst the line of the equatin , so this is the horizontal asymptote.

Example Question #6 : Graphing An Exponential Function

Define functions  and  as follows:

Give the -coordinate of the point of intersection of their graphs.

Possible Answers:

Correct answer:

Explanation:

First, we rewrite both functions with a common base:

 is left as it is.

 can be rewritten as 

To find the point of intersection of the graphs of the functions, set 

The powers are equal and the bases are equal, so we can set the exponents equal to each other and solve:

To find the -coordinate, substitute 4 for  in either definition:

, the correct response.

Example Question #4 : Graphing An Exponential Function

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 #8 : Graphing An Exponential Function

Define functions  and  as follows:

Give the -coordinate of the point of intersection of their graphs.

Possible Answers:

Correct answer:

Explanation:

First, we rewrite both functions with a common base:

 is left as it is.

 can be rewritten as 

To find the point of intersection of the graphs of the functions, set 

Since the powers of the same base are equal, we can set the exponents equal:

Now substitute in either function:

, the correct answer.

 

Example Question #9 : Graphing An Exponential Function

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

Correct answer:

Explanation:

Since the -intercept is the point at which the graph of  intersects the -axis, the -coordinate is 0, and the -coordinate is :

,

The  -intercept is the point .

Example Question #10 : Graphing An Exponential Function

Evaluate .

Possible Answers:

The system has no solution.

Correct answer:

Explanation:

Rewrite the system as 

and substitute  and  for  and , respectively, to form the system

Add both sides:

        

.

Now backsolve:

Now substitute back:

and

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