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.

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 .

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 .

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.

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.

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 .

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.

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.

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 .

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