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

Transformations of Cubic & Quartic Functions

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Transformations of Cubic Functions

Let f(x)=x3f(x)=x^3 be the parent function for a cubic polynomial function.

If f(x)=a(b(x−h))3+kf(x)=a(b(x-h))^3+k, then the following is true:
HorizontalVerticalb<0‾:a<0‾:horizontal reflection about thevertical reflection about they-axisx-axis∣b∣>1‾:∣a∣>1‾:horizontal compression of  1b  unitsvertical expansion of ’a’ units∣b∣<1‾:∣a∣<1‾:horizontal expansion of  1b  unitsvertical compression of ’a’ unitsh>0‾:k>0‾:horizontal translation ’h’ units rightvertical translation ’k units uph<0‾:k<0‾:horizontal translation ’h’ units leftvertical translation ’k’ units down\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)(x,y) is a point on f(x)=x3f(x)=x^3, then the point on the transformed function y=a(b(x−h))3+ky=a(b(x-h))^3+k becomes:
(x, y)→(xb+h, ay+k)(x,~y)\rightarrow\Big(\frac{x}{b}+h,~ay+k\Big)

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Example

Let's graph function f(x)=−2(x+1)3+2f(x)=-2(x+1)^3+2, stating the following:
  • The transformations applied to the parent function
  • Domain & Range
  • End Behaviour
  • xx-Intercepts & Multiplicity

The table of values for the parent function y=x3y=x^3:
xy−2−8−1−1001128\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)

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The transformations applied to the parent function:
Horizontal‾Vertical‾Reflection about the x-axisExpansion by a factor of 2Translation 1 unit leftTranslation 2 units up\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)=x3f(x)=x^3 will transform to (x−1, −2y+2)(x-1,~-2y+2)


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

xy−318−2−4−12001−14\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→∞, y→−∞x\rightarrow\infin,~y\rightarrow-\infin
  • As x→−∞, y→∞x\rightarrow-\infin,~y\rightarrow\infin
xx-Int.: x=0x=0
Multiplicity of xx-int: Odd multiplicity of 3

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Example: Transformations of Cubic Functions

Identify the transformations applied to the function f(x)=−4(−12(x−2))3−1f(x)=-4(-\frac{1}{2}(x-2))^3-1 and graph the function.


Horizontal‾Vertical‾Reflection about the y-axisReflection about the x-axisExpansion by a factor of 2Expansion by a factor of 4Translation 2 units rightTranslation 1 unit down\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:
xy−2−8−1−1001128\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(−12(x−2))3−1f(x)=-4(-\frac{1}{2}(x-2))^3-1 :

xy631432−10−5−2−33\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}
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Transformations of Quartic Functions

Let f(x)=x4f(x)=x^4 be the parent function for a quartic polynomial function.

If f(x)=a(b(x−h))4+kf(x)=a(b(x-h))^4+k, then the following is true:
HorizontalVerticalb<0‾:a<0‾:horizontal reflection about thevertical reflection about they-axisx-axis∣b∣>1‾:∣a∣>1‾:horizontal compression of  1b  unitsvertical expansion of ’a’ units∣b∣<1‾:∣a∣<1‾:horizontal expansion of  1b  unitsvertical compression of ’a’ unitsh>0‾:k>0‾:horizontal translation ’h’ units rightvertical translation ’k units uph<0‾:k<0‾:horizontal translation ’h’ units leftvertical translation ’k’ units down\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)(x,y) is a point on f(x)=x4f(x)=x^4, then the point on the transformed function y=a(b(x−h))4+ky=a(b(x-h))^4+k becomes:
(x, y)→(xb+h, ay+k)(x,~y)\rightarrow\Big(\frac{x}{b}+h,~ay+k\Big)

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Example

Let's graph function f(x)=3(−2x−4)4−2f(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=x4y=x^4:
xy−216−110011216\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,∞)[0,\infin)

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The transformations applied to the parent function:
Horizontal‾Vertical‾Reflection about the y-axisCompression by a factor of  12Expansion by a factor of 3Translation 2 units leftTranslation 2 units down\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)=x4f(x)=x^4 will transform to (−12x−2, 3y−2)(-\frac{1}{2}x-2,~3y-2)

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Sketch & the table of values for f(x)=3(−2x−4)4−2f(x)=3(-2x-4)^4-2:
xy−146−1.51−2−2−2.51−346\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,∞)[-2,\infin)
End Behaviour:
  • As x→∞, y→∞x\rightarrow\infin,~y\rightarrow\infin
  • As x→∞, y→∞x\rightarrow\infin,~y\rightarrow\infin
xx-Int.: x≈−2.45, −1.55x\approx-2.45,~-1.55
Mutliplicity: Odd multiplicity of 1 for x≈−2.45, −1.55x\approx-2.45,~-1.55

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Example: Transformations of Quartic Functions

Identify the transformations applied to the function f(x)=−14(−23(x+1))4+1f(x)=-\frac{1}{4}(-\frac{2}{3}(x+1))^4+1 and graph the function.


Horizontal‾Vertical‾Reflection about the y-axisReflection about the x-axisExpansion by a factor of 32Compression by a factor of 14Translation 1 unit leftTranslation 1 unit up\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:
xy−216−110011216\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)=−14(−23(x+1))4+1f(x)=-\frac{1}{4}(-\frac{2}{3}(x+1))^4+1 :
xy2−31234−11−5234−4−3\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)=−13(13(x+1))3−1f(x)=-\frac{1}{3}(\frac{1}{3}(x+1))^3-1 has been transformed from its parent function f(x)=x3f(x)=x^3.

Choose the correct transformations that have been applied to f(x)f(x) and choose the correct order:
  1. A vertical reflection about the x-axis
  2. A horizontal reflection about the y-axis
  3. A horizontal translation 1 unit left
  4. A horizontal translation 1 unit right
  5. A vertical translation 1 unit up
  6. A vertical translation 1 unit down
  7. A vertical compression by a factor of 13\frac{1}{3}
  8. A vertical expansion by a factor of 3
  9. A horizontal compression by a factor of 13\frac{1}{3}
  10. A horizontal expansion by a factor of 3


Practice: Transformations of Cubic & Quartic Functions

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

Practice: Transformations of Cubic & Quartic Functions

The function f(x)=x4f(x)=x^4 has undergone the following transformations:

Horizontal‾Vertical‾Reflection about the y-axisReflection about the x-axisCompression by a factor of 34Expansion by a factor of 32Translation 4 units leftTranslation 1 unit down\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?
Extra Practice
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
The function f(x)=x3f(x)=x^3 has undergone the following transformations:
Horizontal‾Vertical‾Reflection about the x-axisExpansion by a factor of 107Compression by a factor of 56Translation 0.5 units rightTranslation 4 unit up\begin{array}{l cc l} \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\\\ &&&\text{Reflection about the x-axis}\\\\ \text{Expansion by a factor of}~\frac{10}{7}&&&\text{Compression by a factor of}~\frac{5}{6}\\\\ \text{Translation 0.5 units right}&&&\text{Translation 4 unit up} \end{array}
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
The function f(x)=x4f(x)=x^4 has undergone the following transformations:
Horizontal‾Vertical‾Reflection about the y-axisReflection about the x-axisCompression by a factor of 23Expansion by a factor of 54Translation 2 units rightTranslation 4 units down\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{2}{3}&&&\text{Expansion by a factor of}~\frac{5}{4}\\\\ \text{Translation 2 units right}&&&\text{Translation 4 units down} \end{array}
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
Graph the function f(x)=12(x−3)4−1f(x)=\frac{1}{2}(x-3)^4-1.
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
Graph the function f(x)=−(−x−3)3−5f(x)=-(-x-3)^3-5.
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
The function f(x)=−49(49(5−x))2+4f(x)=-\frac{4}{9}(\frac{4}{9}(5-x))^2+4 has been transformed from its parent function f(x)=x2f(x)=x^2.
Choose the correct transformations that have been applied to f(x)f(x) and choose the correct order:
Transformations of Cubic & Quartic Functions
Practice: Transformations of Cubic & Quartic Functions
The function f(x)=−32(47(x+2))4+1f(x)=-\dfrac{3}{2}\Bigg(\dfrac{4}{7}(x+2)\Bigg)^4+1 has been transformed from its parent function f(x)=x4f(x)=x^4.
Choose the correct transformations that have been applied to f(x)f(x) and choose the correct order: