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 \(g(x)\).
\(\textbf{1)}\) \(f(x)=x^2,\,\,\, g(x)=(x-2)^2+1\)
Translated 2 units right and 1 unit up.
\(\textbf{2)}\) \(f(x)=\sqrt{x},\,\,\, g(x)=3\sqrt{x}\)
Vertical dilation (or stretch) by factor of 3.
\(\textbf{3)}\) \(f(x)=\sqrt[3]{x},\,\,\, g(x)=\sqrt[3]{x+5}-2\)
Translated 5 units left and 2 units down.
\(\textbf{4)}\) \(f(x)=x^3,\,\,\, g(x)=\frac{1}{2}(x+1)^3\)
Vertical compression (or shrink) by factor of \(\frac{1}{2}\) and a horizontal translation 1 unit left.
\(\textbf{5)}\) \(f(x)=|x|,\,\,\, g(x)=2|x+1|\)
Vertical dilation (or stretch) by factor of 2 and a horizontal translation of 1 unit left.
\(\textbf{6)}\) \(f(x)=x^2,\,\,\, g(x)=4(x)^2-3\)
Vertical dilation (or stretch) by factor of 4 followed by a vertical translation of 3 units down.
\(\textbf{7)}\) \(f(x)=\sqrt[3]{x},\,\,\, g(x)=\sqrt[3]{3x}\)
Horizontal compression (or shrink) by factor of \(\frac{1}{3}\)
\(\textbf{8)}\) \(f(x)=\sqrt{x},\,\,\, g(x)=\sqrt{\frac{1}{2}x}\)
Horizontal dilation (or stretch) by factor of 2.
\(\textbf{9)}\) \(f(x)=x^3,\,\,\, g(x)=2(x+1)^3+2\)
Vertical dilation (or stretch) by factor of 2 followed by a translation of 1 unit left and 2 units up.
\(\textbf{10)}\) \(f(x)=|x|,\,\,\, g(x)=|4x|-5\)
Horizontal compression (or shrink) by factor of \(\frac{1}{4}\) and a vertical translation of 5 units down.
Find \(g(x)\) after applying listed transformations to \(f(x)\).
\(\textbf{11)}\) \(f(x)=x^2, \text{Find } g(x) \text{ after…}\)
\(\text{Vertical Stretch by factor of 3}\)
\(\text{Followed by reflection over x-axis}\)
\(\text{Followed by translation left 5 units}\)
\(g(x)=-3\left(x+5\right)^2\)
\(\textbf{12)}\) \(f(x)=\sqrt{x}, \text{Find } g(x) \text{ after…}\)
\(\text{Followed by reflection over x-axis}\)
\(\text{Translate up 2 units}\)
\(g(x)=-\sqrt{x}+2\)
\(\textbf{13)}\) \(f(x)=\sqrt{x}, \text{Find } g(x) \text{ after…}\)
\(\text{Translate up 2 units}\)
\(\text{Followed by reflection over x-axis}\)
\(g(x)=-\sqrt{x}-2\)
\(\textbf{14)}\) \(f(x)=2\sqrt{x+5}-2, \text{Find } g(x) \text{ after…}\)
\(\text{Translation right 6 units}\)
\(\text{Followed by translation up 3 units}\)
\(\text{Followed by vertical stretch by factor of 3}\)
\(g(x)=6\sqrt{x-1}+3\)
See Related Pages\(\)
