### All Advanced Geometry Resources

## Example Questions

### Example Question #1 : How To Graph A Quadratic Function

What are the possible values of if the parabola of the quadratic function is concave upward and does not intersect the -axis?

**Possible Answers:**

The parabola cannot exist for any value of .

**Correct answer:**

The parabola cannot exist for any value of .

If the graph of is concave upward, then .

If the graph does not intersect the -axis, then has no real solution, and the discriminant is negative:

For the parabola to have both characteristics, it must be true that and , but these two events are mutually exclusive. Therefore, the parabola cannot exist.

### Example Question #1 : How To Graph A Quadratic Function

Which of the following equations has as its graph a vertical parabola with line of symmetry ?

**Possible Answers:**

**Correct answer:**

The graph of has as its line of symmetry the vertical line of the equation

Since in each choice, we want to find such that

so the correct choice is .

### Example Question #991 : Gmat Quantitative Reasoning

Which of the following equations has as its graph a concave-right horizontal parabola?

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

A horizontal parabola has as its equation, in standard form,

,

with real, nonzero.

Its orientation depends on the sign of . In the equation of a concave-right parabola, is positive, so the correct choice is .

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

The graphs of the functions and have the same line of symmetry.

If we define , which of the following is a possible definition of ?

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

The graph of a function of the form - a quadratic function - is a vertical parabola with line of symmetry .

The graph of the function therefore has line of symmetry

, or

We examine all four definitions of to find one with this line of symmetry.

:

, or

:

, or

, or

, or

Since the graph of the function has the same line of symmetry as that of the function , that is the correct choice.

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

Give the -coordinate of a point at which the graphs of the equations

and

intersect.

**Possible Answers:**

**Correct answer:**

We can set the two quadratic expressions equal to each other and solve for .

and , so

The -coordinates of the points of intersection are 2 and 6. To find the -coordinates, substitute in either equation:

One point of intersection is .

The other point of intersection is .

1 is not among the choices, but 41 is, so this is the correct response.

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

Give the set of intercepts of the graph of the function .

**Possible Answers:**

**Correct answer:**

The -intercepts, if any exist, can be found by setting :

The only -intercept is .

The -intercept can be found by substituting 0 for :

The -intercept is .

The correct set of intercepts is .

### Example Question #52 : Graphing

Give the -coordinate of a point of intersection of the graphs of the functions

and

.

**Possible Answers:**

**Correct answer:**

The system of equations can be rewritten as

.

We can set the two expressions in equal to each other and solve:

We can substitute back into the equation , and see that either or . The latter value is the correct choice.

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

has as its graph a vertical parabola on the coordinate plane. You are given that and , but you are not given .

Which of the following can you determine without knowing the value of ?

I) Whether the graph is concave upward or concave downward

II) The location of the vertex

III) The location of the -intercept

IV) The locations of the -intercepts, if there are any

V) The equation of the line of symmetry

**Possible Answers:**

III and IV only

I, II, and V only

I and III only

I and V only

I, III, and IV only

**Correct answer:**

I and III only

I) The orientation of the parabola is determined solely by the sign of . Since , the parabola can be determined to be concave downward.

II and V) The -coordinate of the vertex is ; since you are not given , you cannot find this. Also, since the line of symmetry has equation , for the same reason, you cannot find this either.

III) The -intercept is the point at which ; by substitution, it can be found to be at . known to be equal to 9, so the -intercept can be determined to be .

IV) The -intercept(s), if any, are the point(s) at which . This is solvable using the quadratic formula

Since all three of and must be known for this to be evaluated, and only is known, the -intercept(s) cannot be identified.

The correct response is I and III only.

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

Which of the following equations can be graphed with a vertical parabola with exactly one -intercept?

**Possible Answers:**

**Correct answer:**

The graph of has exactly one -intercept if and only if

has exactly one solution - or equivalently, if and only if

Since in all three equations, , we find the value of that makes this statement true by substituting and solving:

The correct choice is .

### Example Question #10 : Graphing A Quadratic Function

has as its graph a vertical parabola on the coordinate plane. You are given that , but you are given neither nor .

Which of the following can you determine without knowing the values of and ?

I) Whether the graph is concave upward or concave downward

II) The location of the vertex

III) The location of the -intercept

IV) The locations of the -intercepts, if there are any

V) The equation of the line of symmetry

**Possible Answers:**

I only

V only

III only

II and V only

III and IV only

**Correct answer:**

I only

I) The orientation of the parabola is determined solely by the value of . Since , the parabola can be determined to be concave upward.

II and V) The -coordinate of the vertex is ; since you are not given , you cannot find this. Also, since the line of symmetry has equation , for the same reason, you cannot find this either.

III) The -intercept is the point at which ; by substitution, it can be found to be at . is unknown, so the -intercept cannot be found.

IV) The -intercept(s), if any, are the point(s) at which . This is solvable using the quadratic formula

Since all three of and must be known for this to be evaluated, and only is known, the -intercept(s) cannot be identified.

The correct response is I only.

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