Transformations of Functions

Notes

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

Practice Problems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

See Related Pages\(\)

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

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

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

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

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