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Notes

\({\text{Transformations of Functions}}\)
\(\underline{\text{Function/Transformation}}\)	\(\underline{\text{Description}}\)
\(f(x)\)	\(\text{Parent Function}\)
\(f(x+h)\)	\(\text{Translated } h \text{ units left}\)
\(f(x-h)\)	\(\text{Translated } h \text{ units right}\)
\(f(x)+k\)	\(\text{Translated } k \text{ units up}\)
\(f(x)-k\)	\(\text{Translated } k \text{ units down}\)
\(a \cdot f(x)\)	\(\text{Vertical dilation (stretch) by a factor of } a \)
\(\frac{1}{a} \cdot f(x)\)	\(\text{Vertical dilation (shrink) by a factor of } \frac{1}{a}\)
\(f(b \cdot x)\)	\(\text{horizontal dilation (shrink) by a factor of } \frac{1}{b} \)
\(f\left(\frac{1}{b} \cdot x\right)\)	\(\text{horizontal dilation (stretch) by a factor of }b \)
\(-f(x)\)	\(\text{Reflection over x-axis}\)
\(f(-x)\)	\(\text{Reflection over y-axis}\)

Practice Problems

Describe the transformations from \(f(x)\) to \(g(x)\).

\(\textbf{1)}\) \(f(x)=x^2,\,\,\, g(x)=(x-2)^2+1\)

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Translated 2 units right and 1 unit up.

\(\textbf{2)}\) \(f(x)=\sqrt{x},\,\,\, g(x)=3\sqrt{x}\)

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Vertical dilation (or stretch) by factor of 3.

\(\textbf{3)}\) \(f(x)=\sqrt[3]{x},\,\,\, g(x)=\sqrt[3]{x+5}-2\)

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Translated 5 units left and 2 units down.

\(\textbf{4)}\) \(f(x)=x^3,\,\,\, g(x)=\frac{1}{2}(x+1)^3\)

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Vertical dilation (or shrink) by factor of \(\frac{1}{2}\) and a horizontal translation 1 unit left.

\(\textbf{5)}\) \(f(x)=|x|,\,\,\, g(x)=2|x+1|\)

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Vertical dilation (or stretch) by factor of 2 and a horizontal translation of 1 unit left.

\(\textbf{6)}\) \(f(x)=x^2,\,\,\, g(x)=4(x)^2-3\)

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Vertical dilation (or stretch) by factor of 4 followed by a vertical translation of 3 units down.

\(\textbf{7)}\) \(f(x)=\sqrt[3]{x},\,\,\, g(x)=\sqrt[3]{3x}\)

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Horizontal dilation (or shrink) by factor of \(\frac{1}{3}\)

\(\textbf{8)}\) \(f(x)=\sqrt{x},\,\,\, g(x)=\sqrt{\frac{1}{2}x}\)

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Horizontal dilation (or stretch) by factor of 2.

\(\textbf{9)}\) \(f(x)=x^3,\,\,\, g(x)=2(x+1)^3+2\)

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Vertical dilation (or stretch) by factor of 2 followed by a translation of 1 unit left and 2 units up.

\(\textbf{10)}\) \(f(x)=|x|,\,\,\, g(x)=|4x|-5\)

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Horizontal dilation (or shrink) by factor of \(\frac{1}{4}\) and a vertical translation of 5 units down.

Find \(g(x)\) after applying listed transformations to \(f(x)\).

\(\textbf{11)}\) \(f(x)=x^2, \text{Find } g(x) \text{ after…}\)
\(\text{Vertical Stretch by factor of 3}\)
\(\text{Followed by reflection over x-axis}\)
\(\text{Followed by translation left 5 units}\)

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\(g(x)=-3\left(x+5\right)^2\)

\(\textbf{12)}\) \(f(x)=\sqrt{x}, \text{Find } g(x) \text{ after…}\)
\(\text{Followed by reflection over x-axis}\)
\(\text{Translate up 2 units}\)

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\(g(x)=-\sqrt{x}+2\)

\(\textbf{13)}\) \(f(x)=\sqrt{x}, \text{Find } g(x) \text{ after…}\)
\(\text{Translate up 2 units}\)
\(\text{Followed by reflection over x-axis}\)

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\(g(x)=-\sqrt{x}-2\)

\(\textbf{14)}\) \(f(x)=2\sqrt{x+5}-2, \text{Find } g(x) \text{ after…}\)
\(\text{Translation right 6 units}\)
\(\text{Followed by translation up 3 units}\)
\(\text{Followed by vertical stretch by factor of 3}\)

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\(g(x)=6\sqrt{x-1}+3\)

See Related Pages\(\)

\(\bullet\text{ Graphing Square Root Functions}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graphing Quadratic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)= f(x)=(x+3)^2-2 \)

\(\bullet\text{ Graphing Cube Root Functions}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graphing Absolute Value Functions}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Andymath Homepage}\)

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