# SAT II Math II : Graphing Quadratic Functions and Conic Sections

## Example Questions

### Example Question #72 : Functions And Graphs

What is the center and radius of the circle indicated by the equation?

Explanation:

A circle is defined by an equation in the format .

The center is indicated by the point  and the radius .

In the equation , the center is  and the radius is .

### Example Question #1 : Graphing Quadratic Functions And Conic Sections

Give the axis of symmetry of the parabola of the equation

Explanation:

The line of symmetry of the parabola of the equation

is the vertical line

Substitute :

The line of symmetry is

That is, the line of the equation .

### Example Question #2 : Graphing Quadratic Functions And Conic Sections

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

and

and

The parabola has no -intercept.

and

and

Explanation:

Set  and solve for :

The terms have a GCF of 2, so

The trinomial in parentheses can be FOILed out by noting that  and :

Set each of the linear binomials to 0 and solve for :

or

The parabola has as its two intercepts the points  and .

### Example Question #3 : Graphing Quadratic Functions And Conic Sections

Give the -coordinate of the vertex of the parabola of the function

Explanation:

The -coordinate of the vertex of a parabola of the form

is

.

Substitute :

The -coordinate is therefore :

### Example Question #4 : Graphing Quadratic Functions And Conic Sections

A baseball is thrown straight up with an initial speed of 60 miles per hour by a man standing on the roof of a 100-foot high building. The height of the baseball in feet is modeled by the function

To the nearest foot, how high is the baseball when it reaches the highest point of its path?

Explanation:

We are seeking the value of  when the graph of  - a parabola - reaches its vertex.

To find this value, we first find the value of . For a parabola of the equation

,

the  value of the vertex is

.

Substitute :

The height of the baseball after 1.875 seconds will be

feet.

### Example Question #5 : Graphing Quadratic Functions And Conic Sections

Give the -coordinate of the vertex of the parabola of the function

.

Explanation:

The -coordinate of the vertex of a parabola of the form

is

.

Set :

The -coordinate is therefore :

, which is the correct choice.

### Example Question #6 : Graphing Quadratic Functions And Conic Sections

A baseball is thrown upward from the top of a one hundred and fifty-foot-high building. The initial speed of the ball is forty-five miles per hour. The height of the ball after  seconds is modeled by the function

How high does the ball get (nearest foot)?

Explanation:

A quadratic function such as  has a parabola as its graph. The high point of the parabola - the vertex - is what we are looking for.

The vertex of a function

has as coordinates

.

The second coordinate is the height and we are looking for this quantity. Since , setting :

seconds for the ball to reach the peak.

The height of the ball at this point is , which can be evaluated by substitution:

Round this to 182 feet.