### All SAT II Math I Resources

## Example Questions

### Example Question #121 : Functions And Graphs

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

**Possible Answers:**

**Correct answer:**

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

is

.

Set :

The -coordinate is therefore :

, which is the correct choice.

### Example Question #1 : Graphing Quadratic Functions

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

**Possible Answers:**

The parabola has no -intercept.

and

and

and

**Correct answer:**

and

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 Quadratic Functions

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

**Possible Answers:**

**Correct answer:**

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 #4 : Graphing Quadratic Functions

Consider the equation:

The vertex of this parabolic function would be located at:

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

down

right

up

left

**Correct answer:**

right

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
- 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 #1 : Graphing Parabolas

Find the vertex form of the following quadratic equation:

**Possible Answers:**

**Correct answer:**

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 Parabolas

**Possible Answers:**

Red line

Green line

Purple line

None of them

Blue line

**Correct answer:**

Red line

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 #1 : Graphing Quadratic Functions

Which of the following parabolas is downward facing?

**Possible Answers:**

**Correct answer:**

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 #1 : Graphing Parabolas

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

**Possible Answers:**

; minimum

; maximum

; minimum

; maximum

**Correct answer:**

; minimum

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?

.

.

.

**Possible Answers:**

only

and

only

, , and

and

**Correct answer:**

, , and

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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