Cubic Function Transformations
Cubic Function Transformation Exercise
Cubic Functions: Transformations and Parameters
Cubic functions are third-degree polynomials, often represented in the transformation form
f(x) = a a(bx - h))3 + k. Unlike linear functions, these form a characteristic "S" shape or "wiggle," showing a changing curvature and featuring exactly one inflection point.
The Impact of Setting Parameters
By setting the specific values for each parameter, we can precisely manipulate the graph's orientation and position on the coordinate plane:
• Vertical Stretch and Reflection (a): By setting the value of a, we determine the vertical steepness and whether the graph is reflected. In the example provided, setting a = —0.6 results in a vertical compression and a reflection, causing the graph to decrease from left to right.
• Horizontal Scaling (b): Setting the parameter b controls the horizontal stretch or compression. Here,
setting b = 1.3 slightly compresses the curve horizontally toward the inflection point.
• Horizontal and Vertical Shifts (h and k): By setting the values for h and k, we define the exact location
of the inflection point. For this specific graph, setting h = 1 and k = 2.6 repositions the center of the
"S" curve to the coordinates (1, 2.6).
Summary of the Transformation
Through the intentional setting of these parameters, the parent function y = x3 (shown in green) is
transformed into the red curve. This illustrates how every coefficient in a cubic equation directly dictates a physical movement or reshaping of the graph, resulting in a smooth, repositioned, and reflected S-shaped curve.
The cubic function is y = x3 , denoted by function g.
The transformed basic function is y = a(bx - h)3 +k
Note: The 'slider' feature on the x-y coordinate plane can be used to change the a, b, h, and k values
for the following exercises. To do so, place the cursor and hold it on the dot of the slider and
slide it to the desired m and b values.
To move the slider to a different location on the x-y plane, place the cursor and hold it on the line
of the slider and move it to the desired location.
Note: You can zoom in or out with the mouse.
Cubic Function Transformations
Exercise 1
Perform the following cubic function transformation:
Vertical shift of 3 units up.
The new function is y=x3 +3 , denoted by function f.
Set a=1. Set b=1.
Set h=0 since there is no horizontal shift
Set k=3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 2
Perform the following cubic function transformation:
Vertical shift of 3 units down.
The new function is y=x3 - 3 , denoted by function f.
Set a=1. Set b=1.
Set h=0 since there is no horizontal shift
Set k= - 3 which represents the vertical shift of 3 units down.
Observe the transformation of the cubic function.
Exercise 3
Perform the following cubic function transformation:
Horizontal shift of 3 units to the right.
The new function is y=(x-3)3 , denoted by function f.
Set a=1. Set b=1.
Set h=3 which represents the horizontal shift of 3 units to the right.
Set k=0 since there is not vertical shift.
Observe the transformation of the cubic function.
Exercise 4
Perform the following cubic function transformation:
Horizontal shift of 3 units to the left.
The new function is y=(x+3)3 , denoted by function f.
Set a=1. Set b=1.
Set h=- 3 which represents the horizontal shift of 3 units to the left.
Set k=0 since there is not vertical shift.
Observe the transformation of the cubic function.
Exercise 5
Perform the following cubic function transformation:
Vertical shift of 3 units up plus a horizontal shift of 3 units to the right.
New function: y = (x-3)3 +3 , denoted by function f.
Set a=1. Set b=1.
Set h=3 which represents the horizontal shift of 3 units to the right.
Set k=3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 6
Perform the following cubic function transformation:
Vertical shift of 3 units down plus a horizontal shift of 3 units to the left.
New function: y = (x+3)3 - 3 , denoted by function f.
Set a=1. Set b=1.
Set h=- 3 which represents the horizontal shift of 3 units to the left.
Set k=- 3 which represents the vertical shift of 3 units down.
Observe the transformation of the cubic function.
Exercise 7
Perform the following cubic function transformation:
Vertical shift of 3 units down plus a horizontal shift of 3 units to the right.
New function: y = (x - 3)3 - 3 , denoted by function f.
Set a=1. Set b=1.
Set h= 3 which represents the horizontal shift of 3 units to the right.
Set k=- 3 which represents the vertical shift of 3 units down.
Observe the transformation of the cubic function.
Exercise 8
Perform the following cubic function transformation:
Vertical shift of 3 units up plus a horizontal shift of 3 units to the left.
New function: y = (x + 3)3 + 3 ,denoted by function f.
Set a=1. Set b=1.
Set h= - 3 which represents the horizontal shift of 3 units to the left.
Set k= 3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 9
Perform the following cubic function transformation:
Vertical stretch by a factor of 3.
New function: y = 3 x3 , denoted by function f.
Set a=3. Set b=1.
Set h= 0 since there is no horizontal shift.
Set k= 0 since there is no vertical shift.
Observe the transformation of the cubic function.
Exercise 10
Perform the following cubic function transformation:
Vertical shrink by a factor of 1/3.
New function: y = 1/3 x3 , denoted by function f.
Set a=1/3. Set b=1.
Set h= - 3 which represents the horizontal shift of 3 units to the left.
Set k= 3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 11
Perform the following cubic function transformation:
Horizontal stretch by a factor of 1/3.
New function: y = (1/3x)3 , denoted by function f.
Set a =1. Set b=1/3.
Set h= 0 since there is no horizontal shift.
Set k= 0 since there is no vertical shift.
Observe the transformation of the cubic function.
Exercise 12
Perform the following cubic function transformation:
Horizontal shrink by a factor of 3
New function: y = (3x) 3 , denoted by function f.
Set a =1. Set b=3.
Set h= 0 since there is no horizontal shift.
Set k= 0 since there is no vertical shift.
Observe the transformation of the cubic function.
Exercise 13
Perform the following cubic function transformation:
Vertical shift of 3 units up plus, horizontal shift of 3 units to the left
and a vertical stretch by a factor of 2.
New function: y = 2(x + 3)3 + 3 , denoted by function f.
Set a=1. Set b = 1.
Set h= - 3 which represents the horizontal shift of 3 units to the left.
Set k= 3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 14
Perform the following cubic function transformation:
Vertical shift of 3 units up plus, horizontal shift of 3 units to the left
and a vertical shrink by a factor of 1/2.
New function: y = 1/2(x + 3)3 + 3 , denoted by function f.
Set a=1. Set b = 1.
Set h= - 3 which represents the horizontal shift of 3 units to the left.
Set k= 3 which represents the vertical shift of 3 units up.
Observe the transformation of the cubic function.
Exercise 15
Perform the following cubic function transformation:
Vertical reflection over the x-axis.
New function: y = - x3 , denoted by function f.
Set a=-1. Set b = 1.
Set h= 0 since there is no horizontal shift.
Set k= 0 since there is no vertical shift.
Observe the transformation of the cubic function.
Exercise 16
Perform the following cubic function transformation:
Reflection over the y-axis.
New function: y = (-x)3 , denoted by function f.
Set a=1. Set b = -1.
Set h= 0 since there is no horizontal shift.
Set k= 0 since there is no vertical shift.
Observe the transformation of the cubic function.
Exercise 17
Repeat this exercise as many times as desired until concept is mastered.
Use different values of a, b, h and k.