# Advanced Geometry : How to graph an exponential function

## Example Questions

### Example Question #1 : Graphing An Exponential Function

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

The graph has no -intercept.

Explanation:

Set  and solve for :

### Example Question #2 : Graphing An Exponential Function

Define a function  as follows:

Give the -intercept of the graph of .

The graph of  has no -intercept.

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 #22 : Graphing

Define a function  as follows:

Give the vertical aysmptote of the graph of .

The graph of  does not have a vertical asymptote.

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 .

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

The -intercept is therefore .

### Example Question #5 : Graphing An Exponential Function

Define a function  as follows:

Give the horizontal aysmptote of the graph of .

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 #981 : Gmat Quantitative Reasoning

Define functions  and  as follows:

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

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 #131 : Exponential And Logarithmic Functions

Define a function  as follows:

Give the -intercept of the graph of .

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.

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:

### Example Question #9 : Graphing An Exponential Function

Define a function  as follows:

Give the -intercept of the graph of .

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 .

The system has no solution.

Explanation:

Rewrite the system as

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