Algebra II : Transformations of Parabolic Functions

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #28 : Parabolic Functions

Consider the following two functions:

 and 

How is the function  shifted compared with ?

Possible Answers:

 units right,  units down

 units left,  units down

 units left,  units up

 units right,  units down

 units left,  units down

Correct answer:

 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 #14 : Functions And Graphs

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

Possible Answers:

None of these graphs are correct.

C

B

Correct answer:
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 #421 : Functions And Graphs

Select the function that accuratley fits the graph shown.

Graph_1

Possible Answers:

Correct answer:

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.

.Parent_parabola

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.

Graph_1

Correct Answer:

Example Question #1 : Transformations Of Parabolic Functions

State the vertex of the following parabola 

Possible Answers:

Correct answer:

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 #2 : Transformations Of Parabolic Functions

Transform the following parabola: .

Shift up  and to the left .

Possible Answers:

Correct answer:

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 #3 : Transformations Of Parabolic Functions

Transform the following parabola .

Move  units to the left.

Move  unit down.

Possible Answers:

Correct answer:

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 #1 : Transformations Of Parabolic Functions

Which function represents  being shifted to the left  ?

Possible Answers:

Correct answer:

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 #5 : 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. 

Possible Answers:

Correct answer:

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 #6 : Transformations Of Parabolic Functions

List the transformations of the following function: 

  

Possible Answers:

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 right 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

 

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 left 2 units

Vertical translation up 5 units

Correct answer:

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 #7 : Transformations Of Parabolic Functions

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

Possible Answers:

Correct answer:

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

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