### All GMAT Math Resources

## Example Questions

### Example Question #23 : Coordinate Geometry

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

**Possible Answers:**

The graph has no -intercept.

**Correct answer:**

Set and solve for :

### Example Question #24 : Coordinate Geometry

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.

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 #25 : Coordinate Geometry

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.

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 #1 : 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:**

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 #2 : Graphing An Exponential Function

Define a function as follows:

Give the horizontal aysmptote of the graph of .

**Possible Answers:**

**Correct answer:**

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 #3 : Graphing An Exponential Function

Define functions and as follows:

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

**Possible Answers:**

**Correct answer:**

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:**

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

The -intercept is the point .

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

Define functions and as follows:

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

**Possible Answers:**

**Correct answer:**

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 #6 : Graphing An Exponential Function

Define a function as follows:

Give the -intercept of the graph of .

**Possible Answers:**

**Correct answer:**

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 #7 : Graphing An Exponential Function

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

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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