### All GMAT Math Resources

## Example Questions

### Example Question #41 : Graphing

The equation of a vertical parabola on the coordinate plane can be written in the form

, real, nonzero.

Is this parabola concave upward or concave downward?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

The parabola is concave upward if and only if , and concave downward if and only if . Therefore, we need to know the sign of to answer the question. Statement 2, but not Statement 1, gives us the value of , the sign of which is positive, so Statement 2 alone, but not Statement 1 alone, tells us the parabola is concave upward.

### Example Question #42 : Graphing

What is the equation of the line of symmetry of a vertical parabola on the coordinate plane?

Statement 1: The vertex of the parabola has -coordinate 7.

Statement 2: The vertex of the parabola has -coordinate 8.

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

**Correct answer:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

The line of symmetry of a vertical parabola with vertex at has as its equation . In other words, the -coordinate, which is given in Statement 1 but not Statement 2, is the one and only thing needed.

### Example Question #43 : Graphing

How many -intercepts does a vertical parabola on the coordinate plane have—zero, one, or two?

Statement 1: The parabola intersects the graph of the equation twice.

Statement 2: The parabola has -intercept .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements are true, and consider these two equations:

Case 1:

The -intercept can be proven to be by substituting 0 for :

We show that the graph intersects the line of equation twice by substituting 3 for :

The points of intersection are .

To find the -intercept(s), if any exist, substitute 0 for :

This has no real solutions, so the parabola has no -intercepts.

Case 2:

The -intercept be proved to be by substituting 0 for :

We show that the graph intersects the line of equation twice by substituting 3 for :

We examine the discriminant:

The discriminant is positive, so there are two real solutions, meaning that there are two points of intersection.

To find the -intercept(s), if any exist, substitute 0 for :

We examine this discriminant:

The discriminant is positive, so there are two real solutions, meaning that there are two -intercepts.

Two parabolas have been identified fitting the main condition and those of both statements, but one has no -intercept and one has two. The two statements together provide insufficient information.

### Example Question #44 : Graphing

What is the equation of the line of symmetry of a vertical parabola on the coordinate plane?

Statement 1: The parabola has one of its two -intercepts at the point .

Statement 2: The -intercept of the parabola is at the origin.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the ques

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.tion.

**Correct answer:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.tion.

The line of symmetry of a vertical parabola is the vertical line passing through the vertex. Each statement alone gives only one point on the graph, neither of which is the vertex, so neither statement alone gives sufficient information.

Now assume both statements to be true. Statement 1 gives one -intercept, ; Statement 2 states that the graph passes through the origin, so it is not only the -intercept, it is also the other -intercept. The -coordinate of the vertex is the arithmetic mean of those of the two -intercepts, so that value is

Only the -coordinate of the vertex is needed to answer the question - we can immediately deduce that the line of symmetry is .

### Example Question #45 : Graphing

The equation of a vertical parabola on the coordinate plane can be written in the form

, real, nonzero.

Is this parabola concave upward or concave downward?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The parabola is concave upward if and only if , and concave downward if and only if . Therefore, we need to know the sign of to answer the question. We show that the two statements together provide insufficient information by examining two equations.

Case 1:

; we check the values of the expressions in both statements:

The conditions of both statements are satisfied; since , the parabola that graphs this function is concave upward.

Case 2:

; we check the values of the expressions in both statements:

The conditions of both statements are satisfied; since , the parabola that graphs this function is concave downward.

### Example Question #46 : Graphing

What is the equation of the line of symmetry of a parabola on the coordinate plane?

Statement 1: The vertex of the parabola has -coordinate .

Statement 2: The vertex of the parabola has -coordinate .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The two statements together give the vertex of the parabola, but no clue is given as to the *orientation* of the parabola - horizontal, vertical, or otherwise. Without that information, or any way to find it, the line of symmetry cannot be determined.

### Example Question #47 : Graphing

The equation of a vertical parabola on the coordinate plane can be written in the form

,

where real, and is a nonzero number.

Is this parabola concave upward or concave downward?

Statement 1: .

Statement 2: The parabola has -intercept .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements are true.

The parabola is concave upward if and only if , and concave downward if and only if . Therefore, we need to know the sign of to answer the question. We show that the two statements together provide insufficient information by examining two equations.

Case 1:

If we substitute 0 for , is easily seen to be equal to 6, so the -intercept is . Also:

Case 2:

If we substitute 0 for , is easily seen to be equal to 6, so the -intercept is . Also:

Each equation satisfies the conditions of both statements, but is positive in one equation and negative in the other. can assume either sign, so the question of the parabola's concavity is unresolved.

### Example Question #48 : Graphing

Graph a line, if possible.

Statement 1: The slope is 4.

Statement 2: The y-intercept is 4.

**Possible Answers:**

**Correct answer:**

Statement 1): The slope is 4.

Write the slope-intercept form, and substitute the slope.

The point and the y-intercept are unknown. Either of these will be needed to solve for the graph of this line.

Statement 1) by itself is not sufficient to graph a line.

Statement 2): The y-intercept is 4.

Substitute the y-intercept into the incomplete formula.

The function can then be graphed on the x-y coordinate plane.

Therefore:

### Example Question #49 : Graphing

Find the graph of .

I) is a linear equation which passes through the point .

II) crosses the y-axis at 1300.

**Possible Answers:**

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

**Correct answer:**

Both statements are needed to answer the question.

Find the graph of .

I) is a linear equation which passes through the point .

II) crosses the y-axis at 1300.

To graph a linear equation, we need some combination of slope, y-intercept, or two points.

Statement I tells us is linear and gives us one point.

Statement II gives us the y-intercept of .

We can use Statement I and Statement II to find the slope of . Then, we can plot the given points and continue the line in either direction to get our graph.

Slope:

Plugging in the provided value of , 1300, we have the equation of the line :

### Example Question #50 : Graphing

Find the graph of the linear function .

I) passes through the points and .

II) intercepts the -axis at .

**Possible Answers:**

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

**Correct answer:**

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Find the graph of the linear function .

I) passes through the points and .

II) intercepts the -axis at .

Using I), we can find the slope of the function, and then we can start at either point and extend the slope in either direction to find our graph:

So, using I) we are able to find the slope, from which we can find our graph

II) gives us one point, but without any more information, we cannot use II) by itself to find the rest of the graph

So:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.