# Algebra II : Quadratic Functions

## Example Questions

### Example Question #11 : Transformations Of Parabolic Functions

Which would be the equation of when reflected over the -axis?

Explanation:

To flip this over the -axis, the sign of x changes.

This entails changing to

.

Perhaps a simpler way to think about this is that the vertex for this parabola is at .

If we flip the equation over the y-axis, it will place the vertex at , making our new equation .

### Example Question #441 : Functions And Graphs

Write the equation for the parabola  shifted 3 units to the right and then reflected across the -axis.

Explanation:

To solve this problem, we could complete the square and shift the equation that way, but the vertex ends up being  so this may not be an ideal method. Instead, we know we're shifting the equation 3 units to the right, so we can just plug in for every appearance of x:

To simplify, first expand

Now we can plug that in and continue simplifying:

distribute the 2; combine -3 and -5

combine like terms -12x and x; 18 and -8

Now we want to flip this over the x-axis, meaning that the y coordinates change sign.

This means we have to multiply everything by -1, or simply change the sign of every term on the right side:

### Example Question #442 : Functions And Graphs

Describe the translation in

from the parent function

.

Down three units, left one unit

Up three units, left one unit

Up three units, right one unit

Down three units, right one unit

Down three units, right one unit

Explanation:

Below is the standard equation for parabolas;

Therefore,

and

thus,

the translation from the parent function is down three units, right one unit.

### Example Question #443 : Functions And Graphs

Given the parabola , what is the new equation if the parabola is shifted left two units, and up four units?

Explanation:

Shifting up and down will result in a change in the y-intercept.

Shifting the parabola to the left two units will change the inner term  to , which will be .

Replace the  quantity with .

The new equation is:

### Example Question #444 : Functions And Graphs

Shift  to the left two units and up two units.  What is the new equation?

Explanation:

Vertical shifts will change the value of the y-intercept.  Since this function is to be shifted up two units, add two to the right side of the equation.

This graph shifted two units to the left indicates that its zeros will also shift to the left two units, which means that the  term will become .

Rewrite the equation and expand the binomials.

The new equation is:

### Example Question #445 : Functions And Graphs

Shift the parabola  three units to the right.  What is the new equation?

Explanation:

Shifting this graph three units to the right means that the x-variable will need to be replaced with .  Rewrite the equation.

Use the FOIL method to simplify the binomial.

Simplify the right side.

The equation becomes:

3 spaces down

3 spaces right

3 spaces left

3 spaces up

3 spaces right

Explanation:

### Which of the below quadratic functions will be the widest?

Explanation:

To determine how "wide" or "skinny" a parabola is, we look at the leading coefficient.

The smaller the fraction, the wider a parabola will be.

The larger the whole number, the skinnier the parabola will be.

This will give us the widest parabola.

### Example Question #448 : Functions And Graphs

Shift  up one unit.  What is the new equation?

Explanation:

Simplify the following equation by using the FOIL method with the binomial.

Simplify all terms in the parentheses.

Replace the term and simplify.

The equation in standard form is:

Since this parabola is shifted up one unit, add one to the y-intercept.

### Example Question #449 : Functions And Graphs

Write the equation for the parabola after it has been reflected over the y-axis, then shifted up 2 and left 4.

Explanation:

First, reflect the equation over the y-axis by switching the sign of x:

Now shift up 2 by adding 2:

Now shift left 4 by adding 4 to x:

first expand

Now multiply

Now plug those back in:

combine like terms