# Algebra II : Transformations of Parabolic Functions

## Example Questions

### Example Question #1 : Transformations Of Parabolic Functions

Consider the following two functions:

and

How is the function  shifted compared with ?

units left,  units up

units right,  units down

units left,  units down

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 ?

None of these graphs are correct.

Explanation:

The function  shifts a function f(x) units to the left. Conversely,  shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function  or .

### Example Question #3 : Transformations Of Parabolic Functions

Select the function that accuratley fits the graph shown.

Explanation:

The parent function of a parabola is where are the vertex.

The original graph of a parabolic (quadratic) function has a vertex at (0,0) and shifts left or right by h units and up or down by k units.

.

This function then shifts 1 unit left, and 4 units down, and the negative in front of the squared term denotes a rotation over the x-axis.

### Example Question #4 : Transformations Of Parabolic Functions

State the vertex of the following parabola

Explanation:

Without doing much work or manipulation of the function, we can use our knowledge of Vertex Form of quadratic functions, which is

with  being the coordinates of the vertex. Knowing this, we can analyze our function  to find the vertex...     vertex: .

Note: This function is simply a transformation of the function

### Example Question #5 : Transformations Of Parabolic Functions

Transform the following parabola: .

Shift up  and to the left .

Explanation:

When transforming paraboloas, to translate up, add to the equation (or add to the Y).

To translate to the left, add to the X.

Don't forget that if you add to the X, then since X is squared, the addition to X must also be squared.

with the shift up 5 becomes: .

Now adding the shift to the left we get:

.

### Example Question #1 : Transformations Of Parabolic Functions

Transform the following parabola .

Move  units to the left.

Move  unit down.

Explanation:

To move unit down, subtract from Y (or from the entire equation) , so subtract 1.

To move unit to the left, add to X (don't forget, that since you are squaring X, you must square the addition as well).

With the move down our equation  becomes: .

Now to move it to the left we get .

### Example Question #7 : Transformations Of Parabolic Functions

Which function represents  being shifted to the left  ?

Explanation:

The parent function for a parabolic function is  where  is the center of the parabola. To shift the parabola left of right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right.

### Example Question #8 : Transformations Of Parabolic Functions

Transformations of Parabolic Functions

Given the function:

write the equation of a new function  that has been translated right 2 spaces and up 4 spaces.

Explanation:

Translations that effect x must be directly connected to x in the function and must also change the sign. So when the function was translated right two spaces, a  must be connected to the x value in the function.

Translation that effect y must be directly connected to the constant in the funtion - so when the function was translated up 4 spaces a +4 must be added to the (-5) in the original function.

When both of these happen in the function the new function must become:

### Example Question #9 : Transformations Of Parabolic Functions

List the transformations of the following function:

Compressed by a factor of 3

Horizontal translation to the left 2 units

Vertical translation down 5 units

Compressed by a factor of 3

Horizontal translation to the left 2 units

Vertical translation up 5 units

Stretched by a factor of 3

Horizontal translation to the right 5 units

Vertical translation up 2 units

Stretched by a factor of 3

Horizontal translation to the left 2 units

Vertical translation up 5 units

Compressed by a factor of 3

Horizontal translation to the right 2 units

Vertical translation up 5 units

Compressed by a factor of 3

Horizontal translation to the left 2 units

Vertical translation up 5 units

Explanation:

Because the parent function is , we can write the general form as:

.

a is the compression or stretch factor.

If , the function compresses or "narrows" by a factor of a.

If , the function stretches or "widens" by a factor of a.

b represents how the function shifts horizontally.

If b is negative, the function shifts to the left b units.

If b is positive, the function shifts to the right b units.

c represents how the function shifts vertically.

If c is positive, the function shifts up c units.

If c is negative, the function shifts down c units.

For our problem, a=3, b=-2, and c=5. (Remember that even though b is negative, the negative from the "general form" makes the sign positive). It follows that we have a compression by a factor of 3, a horizontal shift to the left 2 units, and a vertical shift up 5 units.

### Example Question #10 : Transformations Of Parabolic Functions

Translate the parabola up 6 units and right 3.

Explanation:

To shift up 6 units, just add 6:

To shift to the right 3, subtract 3 from x:

First expand :

now this gives us:

distribute the 2 and the 4:

combine like terms: