Example Questions

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?

left

up

down

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 #1 : 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 : Quadratic Functions

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

Red line

Green line

Purple line

Blue line

None of them

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 #8 : Graphing Quadratic Functions

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 #9 : Graphing Quadratic Functions

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

; minimum

; maximum

; maximum

; minimum

; 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 Parabolas

How many -intercepts does the graph of the function

have?

One

Two

None of these

Zero

Four

Zero

Explanation:

The graph of a quadratic function  has an -intercept at any point  at which , so, first, set the quadratic expression equal to 0:

The number of -intercepts of the graph is equal to the number of real zeroes of the above equation, which can be determined by evaluating the discriminant of the equation, . Set , and evaluate:

The discriminant is negative, so the equation has two solutions, neither of which are real. Consequently, the graph of the function  has no -intercepts.

Example Question #1 : Graphing Parabolas

Which of the following graphs matches the function ?

Explanation:

Start by visualizing the graph associated with the function :

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of  looks like this:

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function  :

Example Question #1 : Transformations Of Parabolic Functions

Consider the following two functions:

and

How is the function  shifted compared with ?

units right,  units down

units left,  units down

units left,  units up

units left,  units down

units right,  units down

units left,  units down

Explanation:

The  portion results in the graph being shifted 3 units to the left, while the  results in the graph being shifted six units down.  Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

Example Question #1 : Transformations Of Parabolic Functions

If the function  is depicted here, which answer choice graphs ?