 # Transformations of Cubic Functions

Lesson

## Translating, enlarging and reflecting $y=ax^3$y=ax3

Recall that the cubic function is given by $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d

Now if $b^2=3ac$b2=3ac, the function can be re-written in the form $y=a\left(x-h\right)^3+k$y=a(xh)3+k.

This second form can be considered as the translated form of the cubic given by $y=ax^3$y=ax3. Specifically, if the curve of the function $y=ax^3$y=ax3 is translated (shifted in position without distorting the shape) $h$h units to the right and $k$k units up, then the translated function becomes $y=a\left(x-h\right)^3+k$y=a(xh)3+k.

That is, if we wish to translate a cubic function $y=ax^3$y=ax3 so that its central point moves from the origin to any other point $\left(h,k\right)$(h,k) in the cartesian plane, then all we need to do is to replace $x$x with $\left(x-h\right)$(xh) and $y$y with $\left(y-k\right)$(yk)

As an example, suppose we wish to translate $y=2x^3$y=2x3 so that its central point is $\left(-3,1\right)$(3,1) then we simply change $x$x to $x-\left(-3\right)=x+3$x(3)=x+3 and $y$y to $y-\left(1\right)=y-1$y(1)=y1 so that the translated function becomes $y-1=2\left(x+3\right)^3$y1=2(x+3)3, which when written explicitly becomes $y=2\left(x+3\right)^3+1$y=2(x+3)3+1. It is as simple as that!

The simpler function $y=ax^3$y=ax3 for $\left|a\right|\ne1$|a|1 can be thought of as the enlargement (a stretch or compression in the $y$y direction) of the base cubic function $y=x^3$y=x3

When $a<0$a<0 the base function $y=ax^3$y=ax3 is a reflected version of the same base function $y=\left|a\right|x^3$y=|a|x3. As an example, the function $y=-2x^3$y=2x3 is a reflected version (reflected in the $x$x axis) of $y=2x^3$y=2x3.

## Using the Applet

##### Exercise 1:

Open up the applet below and you will see the graph of $y=x^3$y=x3, written as $y=1\left(x-0\right)^3+0$y=1(x0)3+0. Click on the box marked "show base function", and then, without changing anything else, move the 'a' slider along to $2$2, so that the function then reads $y=2\left(x-0\right)^3+0$y=2(x0)3+0. Notice how the curve narrows, simply because its rising at a faster rate.

Now move the a slider back to $-2$2. Watch how the cubic flips, so that it becomes a reflected image.

##### Exercise 2:

First move the $a$a slider back to $1$1.

Then, for the translation, move the $h$h slider to $-3$3 and the $k$k slider to $-1$1, so that the function then reads $y=1\left(x--3\right)^3+-1$y=1(x3)3+1 .What the function has changed to is, of course, $y=\left(x+3\right)^3-1$y=(x+3)31 so that the central point has shifted from the origin to the point $\left(-3,-1\right)$(3,1).

Finally move the a slider up and down from $2$2 to $-2$2 to see the independent effect of the enlargement. Note that the central point remains the same. This is a critical understanding.

Very importantly, note that the presence of a negative $a$a value shows a reflected cubic function in the elevated line $y=k$y=k (and not the $x$x - axis as it was for the base function).

##### Exercise 3:

Work out what you must slide to show a cubic equation that has a central point at $\left(-3,2\right)$(3,2) and a enlargement factor $a=-\frac{1}{2}$a=12. Try to describe what you see.

#### Worked Examples

##### QUESTION 1

Consider the function $y=\frac{1}{2}\left(x-3\right)^3$y=12(x3)3

1. Is the cubic increasing or decreasing from left to right?

Increasing

A

Decreasing

B

Increasing

A

Decreasing

B
2. Is the function more or less steep than the function $y=x^3$y=x3 ?

More steep

A

Less steep

B

More steep

A

Less steep

B
3. What are the coordinates of the point of inflection of the function?

Inflection ($\editable{}$, $\editable{}$)

4. Plot the graph $y=\frac{1}{2}\left(x-3\right)^3$y=12(x3)3

##### QUESTION 2

Consider the function $y=-2\left(x-2\right)^3$y=2(x2)3

1. Is the cubic increasing or decreasing from left to right?

Increasing

A

Decreasing

B

Increasing

A

Decreasing

B
2. Is the function more or less steep than the function $y=-x^3$y=x3 ?

More steep

A

Less steep

B

More steep

A

Less steep

B
3. What are the coordinates of the point of inflection of the function?

Inflection ($\editable{}$, $\editable{}$)

4. Plot the graph $y=-2\left(x-2\right)^3$y=2(x2)3

##### QUESTION 3

This is a graph of $y=x^3$y=x3.

1. How do we shift the graph of $y=x^3$y=x3 to get the graph of $y=\left(x-2\right)^3-3$y=(x2)33?

Move the graph to the right by $2$2 units and down by $3$3 units.

A

Move the graph to the left by $3$3 units and down by $2$2 units.

B

Move the graph to the right by $3$3 units and up by $2$2 units.

C

Move the graph to the left by $2$2 units and up by $3$3 units.

D

Move the graph to the right by $2$2 units and down by $3$3 units.

A

Move the graph to the left by $3$3 units and down by $2$2 units.

B

Move the graph to the right by $3$3 units and up by $2$2 units.

C

Move the graph to the left by $2$2 units and up by $3$3 units.

D
2. Hence plot $y=\left(x-2\right)^3-3$y=(x2)33 on the same graph as $y=x^3$y=x3.