Wize High School Grade 12 Pre-Calculus Textbook > Polynomial Functions

Transformations of Cubic & Quartic Functions

Transformations of Cubic Functions

Let $f(x)=x^3$ be the parent function for a cubic polynomial function.

If $f(x)=a(b(x-h))^3+k$ , then the following is true:

\begin{array}{|l |c cl|} \hline \text{Horizontal}&&&\text{Vertical}\\ \hline \\ \underline{b<0}:&&&\underline{a<0}:\\ \text{horizontal reflection about the}&&&\text{vertical reflection about the}\\ \text{y-axis}&&&\text{x-axis}\\\\ \hline\\ \underline{|b|>1}:&&&\underline{|a|>1}:\\ \text{horizontal compression of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical expansion of 'a' units}\\\\ \hline\\ \underline{|b|<1}:&&&\underline{|a|<1}:\\ \text{horizontal expansion of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical compression of 'a' units}\\\\ \hline\\ \underline{h>0}:&&&\underline{k>0}:\\ \text{horizontal translation 'h' units right}&&&\text{vertical translation 'k units up}\\\\ \hline\\ \underline{h<0}:&&&\underline{k<0}:\\ \text{horizontal translation 'h' units left}&&&\text{vertical translation 'k' units down}\\ \\ \hline \end{array}

Point Mapping

If the point $(x,y)$ is a point on $f(x)=x^3$ , then the point on the transformed function $y=a(b(x-h))^3+k$ becomes:

(x,~y)\rightarrow\Big(\frac{x}{b}+h,~ay+k\Big)

Example

Let's graph function $f(x)=-2(x+1)^3+2$ , stating the following:

The transformations applied to the parent function
Domain & Range
End Behaviour
$x$ -Intercepts & Multiplicity

The table of values for the parent function $y=x^3$ :

\begin{array}{|c|c|} \hline x&y\\\hline -2&-8\\\hline -1&-1\\\hline 0&0\\\hline 1&1\\\hline 2&8\\\hline \end{array}

Domain: $(-\infin,\infin)$

Range: $(-\infin,\infin)$

The transformations applied to the parent function:

\begin{array}{l c c c l} \underline{\text{Horizontal}}&&&&\underline{\text{Vertical}}\\\\ &&&&\text{Reflection about the x-axis}\\\\ &&&&\text{Expansion by a factor of 2}\\\\ \text{Translation 1 unit left}&&&&\text{Translation 2 units up} \end{array}

Therefore, every point (x, y) on $f(x)=x^3$ will transform to $(x-1,~-2y+2)$

Sketch & the table of values for $f(x)=-2(x+1)^3+2$ :

Image illustrating Transformations of Cubic $\begin{array}{|c|c|} \hline x&y\\\hline -3&18\\\hline -2&-4\\\hline -1&2\\\hline 0&0\\\hline 1&-14\\\hline \end{array}$

Domain: $(-\infin,\infin)$

Range: $(-\infin,\infin)$

End Behaviour:

As $x\rightarrow\infin,~y\rightarrow-\infin$
As $x\rightarrow-\infin,~y\rightarrow\infin$

$x$ -Int.: $x=0$

Multiplicity of $x$ -int: Odd multiplicity of 3

Example: Transformations of Cubic Functions

Identify the transformations applied to the function $f(x)=-4(-\frac{1}{2}(x-2))^3-1$ and graph the function.

\begin{array}{|l c c |l|} \hline \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\\hline\\ \text{Reflection about the y-axis}&&&\text{Reflection about the x-axis}\\\\ \text{Expansion by a factor of 2}&&&\text{Expansion by a factor of 4}\\\\ \text{Translation 2 units right}&&&\text{Translation 1 unit down}\\ \hline \end{array}

The table of values for the parent function is:

\begin{array}{|c|c|} \hline x&y\\\hline -2&-8\\\hline -1&-1\\\hline 0&0\\\hline 1&1\\\hline 2&8\\\hline \end{array}

The table of values and graph of $f(x)=-4(-\frac{1}{2}(x-2))^3-1$ :

Image illustrating Example $\begin{array}{|c|c|} \hline x&y\\\hline 6&31\\\hline 4&3\\\hline 2&-1\\\hline 0&-5\\\hline -2&-33\\\hline \end{array}$

Transformations of Quartic Functions

Let $f(x)=x^4$ be the parent function for a quartic polynomial function.

If $f(x)=a(b(x-h))^4+k$ , then the following is true:

\begin{array}{|l c c |l|} \hline \text{Horizontal}&&&\text{Vertical}\\ \hline\\ \underline{b<0}:&&&\underline{a<0}:\\ \text{horizontal reflection about the}&&&\text{vertical reflection about the}\\ \text{y-axis}&&&\text{x-axis}\\\\ \hline\\ \underline{|b|>1}:&&&\underline{|a|>1}:\\ \text{horizontal compression of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical expansion of 'a' units}\\\\ \hline\\ \underline{|b|<1}:&&&\underline{|a|<1}:\\ \text{horizontal expansion of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical compression of 'a' units}\\\\ \hline\\ \underline{h>0}:&&&\underline{k>0}:\\ \text{horizontal translation 'h' units right}&&&\text{vertical translation 'k units up}\\\\ \hline\\ \underline{h<0}:&&&\underline{k<0}:\\ \text{horizontal translation 'h' units left}&&&\text{vertical translation 'k' units down}\\ \\ \hline \end{array}

Point Mapping

If the point $(x,y)$ is a point on $f(x)=x^4$ , then the point on the transformed function $y=a(b(x-h))^4+k$ becomes:

(x,~y)\rightarrow\Big(\frac{x}{b}+h,~ay+k\Big)

Example

Let's graph function $f(x)=3(-2x-4)^4-2$ , stating the following:

The transformations applied to the parent function
Domain & Range
End Behaviour
X-Intercepts & Multiplicity

The table of values for the parent function $y=x^4$ :

\begin{array}{|c|c|} \hline x&y\\\hline -2&16\\\hline -1&1\\\hline 0&0\\\hline 1&1\\\hline 2&16\\\hline \end{array}

Domain: $(-\infin,\infin)$

Range: $[0,\infin)$

The transformations applied to the parent function:

\begin{array}{l c c c l} \underline{\text{Horizontal}}&&&&\underline{\text{Vertical}}\\\\ \text{Reflection about the y-axis}&&&&\\\\ \text{Compression by a factor of}~~\displaystyle{\frac{1}{2}}&&&&\text{Expansion by a factor of 3}\\\\ \text{Translation 2 units left}&&&&\text{Translation 2 units down} \end{array}

Therefore, every point (x, y) on $f(x)=x^4$ will transform to $(-\frac{1}{2}x-2,~3y-2)$

Sketch & the table of values for $f(x)=3(-2x-4)^4-2$ :

Image illustrating Transformations of Quartic Functions $\begin{array}{|c|c|} \hline x&y\\\hline -1&46\\\hline -1.5&1\\\hline -2&-2\\\hline -2.5&1\\\hline -3&46\\\hline \end{array}$

Domain: $(-\infin,\infin)$

Range: $[-2,\infin)$

End Behaviour:

As $x\rightarrow\infin,~y\rightarrow\infin$
As $x\rightarrow\infin,~y\rightarrow\infin$

$x$ -Int.: $x\approx-2.45,~-1.55$

Mutliplicity: Odd multiplicity of 1 for $x\approx-2.45,~-1.55$

Example: Transformations of Quartic Functions

Identify the transformations applied to the function $f(x)=-\frac{1}{4}(-\frac{2}{3}(x+1))^4+1$ and graph the function.

\begin{array}{l c c l} \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\\\ \text{Reflection about the y-axis}&&&\text{Reflection about the x-axis}\\\\ \text{Expansion by a factor of}~\frac{3}{2}&&&\text{Compression by a factor of}~\frac{1}{4}\\\\ \text{Translation 1 unit left}&&&\text{Translation 1 unit up} \end{array}

The table of values for the parent function is:

\begin{array}{|c|c|} \hline x&y\\\hline -2&16\\\hline -1&1\\\hline 0&0\\\hline 1&1\\\hline 2&16\\\hline \end{array}

The table of values and graph of $f(x)=-\frac{1}{4}(-\frac{2}{3}(x+1))^4+1$ :

Image illustrating Example $\begin{array}{|c|c|} \hline x&y\\\hline 2&-3\\\hline \frac{1}{2}&\frac{3}{4}\\\hline -1&1\\\hline -\frac{5}{2}&\frac{3}{4}\\\hline -4&-3\\\hline \end{array}$

Practice: Transformations of Cubic & Quartic Functions

The function $f(x)=-\frac{1}{3}(\frac{1}{3}(x+1))^3-1$ has been transformed from its parent function $f(x)=x^3$ .

Choose the correct transformations that have been applied to $f(x)$ and choose the correct order:

A vertical reflection about the x-axis
A horizontal reflection about the y-axis
A horizontal translation 1 unit left
A horizontal translation 1 unit right
A vertical translation 1 unit up
A vertical translation 1 unit down
A vertical compression by a factor of $\frac{1}{3}$
A vertical expansion by a factor of 3
A horizontal compression by a factor of $\frac{1}{3}$
A horizontal expansion by a factor of 3

I don't know

Practice: Transformations of Cubic & Quartic Functions

Graph the function $f(x)=-(2x+4)^3-2$ .

Illustration for practice question: Transformations of Cubic & Quartic Functions

I don't know

Practice: Transformations of Cubic & Quartic Functions

The function $f(x)=x^4$ has undergone the following transformations:

\begin{array}{l cc l} \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\\\ \text{Reflection about the y-axis}&&&\text{Reflection about the x-axis}\\\\ \text{Compression by a factor of}~\frac{3}{4}&&&\text{Expansion by a factor of}~\frac{3}{2}\\\\ \text{Translation 4 units left}&&&\text{Translation 1 unit down} \end{array}

What is a function that represents the transformation?

Your answer

I don't know