# SAT II Math I : Graphing Quadratic Functions

## Example Questions

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### Example Question #1 : Graphing Quadratic Functions

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 #2 : Graphing Quadratic Functions

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

and

and

and

The parabola has no -intercept.

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 #1 : Graphing Parabolas

All of the following are equations of down-facing parabolas EXCEPT:

Explanation:

A parabola that opens downward has the general formula

,

as the negative sign in front of the  term makes flips the parabola about the horizontal axis.

By contrast, a parabola of the form rotates about the vertical axis, not the horizontal axis.

Therefore, is not the equation for a parabola that opens downward.

### Example Question #1 : Graphing Quadratic Functions

Consider the equation:

The vertex of this parabolic function would be located at:

Explanation:

For any parabola, the general equation is

, and the x-coordinate of its vertex is given by

.

For the given problem, the x-coordinate is

.

To find the y-coordinate, plug  into the original equation:

Therefore the vertex is at .

### Example Question #1 : Graphing Parabolas

In which direction does graph of the parabola described by the above equation open?

down

left

up

right

right

Explanation:

Parabolas can either be in the form

for vertical parabolas or in the form

for horizontal parabolas. Since the equation that the problem gives us has a y-squared term, but not an x-squared term, we know this is a horizontal parabola. The rules for a horizontal parabola are as follows:

• If , then the horizontal parabola opens to the right.
• If , then the horizontal parabola opens to the left.

In this case, the coefficient in front of the y-squared term is going to be positive, once we isolate x. That makes this a horizontal parabola that opens to the right.

### Example Question #2 : Graphing Quadratic Functions

Find the vertex form of the following quadratic equation:

Explanation:

Factor 2 as GCF from the first two terms giving us:

Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because ) resulting in the following equation:

which is equal to

Hence the vertex is located at

### Example Question #1 : Graphing Quadratic Functions

Based on the figure below, which line depicts a quadratic function?

Green line

Blue line

Purple line

None of them

Red line

Red line

Explanation:

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to .

The blue line represents a linear function and will have a formula similar to .

The green line represents an exponential function and will have a formula similar to .

The purple line represents an absolute value function and will have a formula similar to .

### Example Question #5 : Graphing Parabolas

Which of the following parabolas is downward facing?

Explanation:

We can determine if a parabola is upward or downward facing by looking at the coefficient of the  term. It will be downward facing if and only if this coefficient is negative. Be careful about the answer choice . Recall that this means that the entire value inside the parentheses will be squared. And, a negative times a negative yields a positive. Thus, this is equivalent to . Therefore, our answer has to be

### Example Question #3 : Graphing Quadratic Functions

What is the vertex of the function ? Is it a maximum or minimum?

; maximum

; minimum

; minimum

; maximum

; minimum

Explanation:

The equation of a parabola can be written in vertex form: .

The point  in this format is the vertex. If  is a postive number the vertex is a minimum, and if  is a negative number the vertex is a maximum.

In this example, . The positive value means the vertex is a minimum.

### Example Question #1 : Graphing Quadratic Functions

How many points of intersection could two distinct quadratic functions have?

.

, and

only

only

and

and

, and

Explanation:

An intersection of two functions is a point they share in common. A diagram can show all the possible solutions:

Notice that:

and  intersect  times

and  intersect  time

and  intersect  times

The diagram shows that , , and  are all possible.

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