### All GMAT Math Resources

## Example Questions

### Example Question #1 : Dsq: Graphing A Quadratic Function

The graph of the function is a parabola. Is this parabola concave upward or is it concave downward?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is 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.

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

**Correct answer:**

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

Whether the parabola of a quadratic function is concave upward or concave downward depends on one thing and one thing only - whether quadratic coefficient is positive or negative. Statement 1 gives you this information; Statement 2 does not.

### Example Question #1 : Dsq: Graphing A Quadratic Function

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

Statement 1: The -intercept of the parabola is .

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

**Possible Answers:**

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

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

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient 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 line of symmetry of a vertical parabola is the vertical line passing through the vertex. Statement 1 alone is not helpful, since it only gives the -intercept.

Statement 2 alone, however, answers the question. In a parabola with only one -intercept, that -intercept, given in Statement 2 as , doubles as the vertex. The vertical line through the vertex, which here is the line with equation , is the line of symmetry.

### Example Question #1 : Dsq: Graphing A Quadratic Function

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

, real, nonzero.

How many -intercepts does the parabola have - zero, one, or two?

Statement 1:

Statement 2:

**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:**

Assume Statement 1 alone. The number of -intercepts of the graph of the function - depends on the sign of the discriminant of the expression, .

If , then the discriminant becomes

Since in a quadratic equation, is nonzero, must be positive, and discriminant must be negative. This means that the parabola of has no -intercepts.

We show that Statement 2 alone gives insufficient information by examining two equations: and . In both equations, the sum of the coefficients is 9.

In the first equation, the discriminant is

, a positive value, so the parabola of has two -intercepts.

In the second equation, however, the discriminant is

, a negative value, so the parabola of has no -intercepts.

### Example Question #4 : Dsq: Graphing A Quadratic Function

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

Statement 1: The parabola passes through points and .

Statement 2: The parabola passes through the points and .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. By vertical symmetry, if two points of a parabola have the same -coordinate, the line of symmetry is the vertical line that passes halfway between them. and have the same -coordinate, so the axis of symmetry must be

, or .

Statement 1 alone is sufficient.

Statement 2 can be proved sufficient using a similar argument.

### Example Question #5 : Dsq: Graphing A Quadratic Function

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

Statement 1: The vertex of the parabola has -coordinate 4.

Statement 2: The vertex of the parabola has -coordinate 9.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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

### Example Question #6 : Dsq: Graphing A Quadratic Function

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: 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:**

Assume Statement 1 alone. Since - that is, the function has a negative discriminant - the graph of has no -intercepts. This alone, however, does not determine whether the parabola is concave upward or concave downward. Also, Statement 2 alone only gives one point of the parabola, thereby providing insufficient information.

Now assume both statements are true. From Statement 2, , so the parabola has a point above the -axis. If the parabola is concave downward, then it must cross the -axis, which is impossible as a result of Statement 1. The parabola therefore must be concave upward.

### Example Question #7 : Dsq: Graphing A Quadratic Function

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

,

where are real, and is a nonzero number.

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

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. The number of -intercepts(s) of the graph of depends on the sign of discriminant . By Statement 1, , or, equivalently, , which means that the parabola of has exactly one -intercept.

Statement 2 alone, that the quadratic coefficient is positive, only establishes that the parabola is concave upward. Therefore, it gives insufficient information.

### Example Question #8 : Dsq: Graphing A Quadratic Function

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

Statement 1: The vertex of the parabola is .

Statement 2: The -intercept of the parabola is .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

From Statement 1, since the vertex is not on the -axis - its -coordinate is nonzero - the parabola has either zero or two -intercepts. However, with no further information, it is not possible to choose. Statement 2 alone is not helpful since it only gives one point, and no further information about it.

Assume both statements to be true. We can find the equation of the parabola as follows:

A parabola with vertex has equation

for some nonzero .

From Statement 1, , so the equation becomes

Since the parabola passes through To find , we substitute 0 for and 21 for :

The equation of the parabola is .

Now that the equation is known, the -intercept(s) themselves, if any, can be found by substituting 0 for .

### Example Question #9 : Dsq: Graphing A Quadratic Function

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 insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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 #10 : Dsq: Graphing A Quadratic Function

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:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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.

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