# Math Tip of the Week: Parent Functions and Transformations

This topic is a popular one; it seems like at the beginning of most Algebra II and Pre-Calculus courses, parent functions and their transformations are discussed.

Here’s my table of the “most popular” parent functions and their graphs:

 Parent Function Graph Parent Function Graph $$y=x$$ Linear, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$   End Behavior**: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)$$  $$y=\left| x \right|$$ Absolute Value, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)$$  $$y={{x}^{2}}$$ Quadratic, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)$$  $$y=\sqrt{x}$$ Radical (Square Root), Neither Domain: $$\left[ {0,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$   End Behavior: $$\displaystyle \begin{array}{l}x\to 0,\,\,\,\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {0,\,0} \right),\,\left( {1,\,1} \right),\,\left( {4,\,2} \right)$$  $$y={{x}^{3}}$$ Cubic, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)$$  $$y=\sqrt{x}$$ Cube Root, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)$$  $$\begin{array}{c}y={{b}^{x}},\,\,\,b>1\,\\(y={{2}^{x}})\end{array}$$ Exponential, Neither Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {0,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {-1,\,\frac{1}{b}} \right),\,\left( {0,\,1} \right),\,\left( {1,\,b} \right)$$ Asymptote:  $$y=0$$  $$\begin{array}{c}y={{\log }_{b}}\left( x \right),\,\,b>1\,\,\,\\(y={{\log }_{2}}x)\end{array}$$ Log, Neither Domain: $$\left( {0,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to {{0}^{+}}\text{, }\,y\to -\infty \\x\to \infty \text{, }\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \left( {\frac{1}{b},\,-1} \right),\,\left( {1,\,0} \right),\,\left( {b,\,1} \right)$$ Asymptote: $$x=0$$  $$\displaystyle y=\frac{1}{x}$$ Rational (Inverse), Odd Domain: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$ Range: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}$$ Critical points: $$\displaystyle \left( {-1,\,-1} \right),\,\left( {1,\,1} \right)$$ Asymptotes: $$y=0,\,\,x=0$$  $$\displaystyle y=\frac{1}{{{{x}^{2}}}}$$ Rational (Inverse Squared),          Even Domain: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$ Range: $$\left( {0,\infty } \right)$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}$$ Critical points: $$\displaystyle \left( {-1,\,1} \right),\,\left( {1,\,1} \right)$$ Asymptotes: $$x=0,\,\,y=0$$  $$y=\text{int}\left( x \right)=\left\lfloor x \right\rfloor$$ Greatest Integer*, Neither Domain:$$\left( {-\infty ,\infty } \right)$$ Range: $$\{y:y\in \mathbb{Z}\}\text{ (integers)}$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Critical points: $$\displaystyle \begin{array}{l}x:\left[ {-1,\,0} \right)\,\,\,y:-1\\x:\left[ {0,\,1} \right)\,\,\,y:0\\x:\left[ {1,\,2} \right)\,\,\,y:1\end{array}$$  $$y=C$$   ($$y=2$$) Constant, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\{y:y=C\}$$   End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to C\\x\to \infty \text{, }\,\,\,y\to C\end{array}$$ Critical points: $$\displaystyle \left( {-1,\,C} \right),\,\left( {0,\,C} \right),\,\left( {1,\,C} \right)$$  When functions are transformed on the outside of the $$f(x)$$ part, you move the function up and down and do the “regular” math, as we’ll see in the examples below.  These are vertical transformations.

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Transformation What It Does Example Graph

$$f\left( x \right)+b$$

Translation

Move graph up  $$b$$  units

Every point on the graph is shifted up  $$b$$  units.

The $$x$$’s stay the same; add  $$b$$  to the $$y$$ values.

Parent:

$$y={{x}^{2}}$$

Transformed: $$y={{x}^{2}}+ \,2$$

 x y  y+2 –1 1    3 0 0    2 1 1    3 Domain:  $$\left( {-\infty ,\infty } \right)$$     Range:  $$\left[ {2,\,\,\infty } \right)$$

$$f\left( x \right)-b$$

Translation

Move graph down  $$b$$  units

Every point on the graph is shifted down  $$b$$  units.

The $$x$$’s stay the same; subtract  $$b$$  from the $$y$$ values.

Parent:

$$y=\sqrt{x}$$

Transformed:

$$y=\sqrt{x}- \,3$$

 x y   y–3 0 0   –3 1 1   –2 4 2   –1 Domain:  $$\left[ {0,\,\,\infty } \right)$$     Range: $$\left[ {-3,\,\,\infty } \right)$$

$$a\,\cdot f\left( x \right)$$

Dilation

Stretch graph vertically by a scale factor of  $$a$$ (sometimes called a dilation).

Note that if $$a<1$$, the graph is compressed or shrunk.

Every point on the graph is stretched  $$a$$  units.

The $$x$$’s stay the same; multiply the $$y$$ values by $$a$$.

Parent:

$$y={{x}^{3}}$$

Transformed:

$$y={{4x}^{3}}$$

 x y   4y –1 –1  –4 0 0    0 1 1    4 Domain: $$\left( {-\infty ,\infty } \right)$$     Range: $$\left( {-\infty ,\infty } \right)$$

$$-f\left( x \right)$$

Reflection

Flip graph around the $$x$$-axis.

Every point on the graph is flipped vertically.

The $$x$$’s stay the same; multiply the $$y$$ values by $$-1$$.

Parent:

$$y=\left| x \right|$$

Transformed:

$$y=-\left| x \right|$$

 x y   –y –1 1   –1 0 0     0 1 1   –1 Domain: $$\left( {-\infty ,\infty } \right)$$     Range: $$\left( {-\infty ,\,\,0} \right]$$

$$\left| {f\left( x \right)} \right|$$

Absolute Value on the $$y$$

Reflect part of graph underneath the $$x$$-axis (negative $$y$$’s) across the $$x$$-axis. Leave positive $$y$$’s the same.

The $$x$$’s stay the same; take the absolute value of the $$y$$’s.

Parent:  $$y=\sqrt{x}$$

Transformed: $$y=\left| {\sqrt{x}} \right|$$

 x y   |y| –1 –1    1 0 0      0 1 1      1 Domain:  $$\left( {-\infty ,\infty } \right)$$     Range:  $$\left[ {0,\infty } \right)$$

When transformations are made on the inside of the $$f(x)$$ part, you move the function back and forth (but do the “opposite” math – basically since if you were to isolate the x, you’d move everything to the other side).  These are horizontal translations.

Transformation What It Does Example Graph
$$f\left( {x+b} \right)$$

Translation

Move graph left  $$b$$  units

(Do the “opposite” when change is inside the parentheses or underneath radical sign.)

Every point on the graph is shifted left  $$b$$  units.

The $$y$$’s stay the same; subtract  $$b$$  from the $$x$$ values.

Parent:

$$y={{x}^{2}}$$

Transformed:

$$y={{\left( {x+2} \right)}^{2}}$$

 x – 2    x y –3    –1 1 –2      0 0 –1      1 1 Domain: $$\left( {-\infty ,\infty } \right)$$     Range: $$\left[ {0,\,\,\infty } \right)$$

$$f\left( {x-b} \right)$$

Translation

Move graph right  $$b$$  units

Every point on the graph is shifted right  $$b$$  units.

The $$y$$’s stay the same; add  $$b$$  to the $$x$$ values.

Parent:

$$y=\sqrt{x}$$

Transformed:

$$y=\sqrt{{x- \,3}}$$

 x + 3   x y 3      0 0 4      1 1 7      4 2 Domain: $$\left[ {-3,\infty } \right)$$      Range: $$\left[ {0,\,\,\infty } \right)$$

$$f\left( {a\cdot x} \right)$$

Dialation

Compress graph horizontally by a scale factor of  $$a$$  units (stretch or multiply by $$\displaystyle \frac{1}{a}$$)

Every point on the graph is compressed  $$a$$  units horizontally.

The $$y$$’s stay the same; multiply the $$x$$ values by $$\displaystyle \frac{1}{a}$$.

Parent:

$$y={{x}^{3}}$$

Transformed:

$$y={{\left( {4x} \right)}^{3}}$$

 $$\frac{1}{4}x$$    x y $$-\frac{1}{4}$$    –1 –1 0      0 0 $$\frac{1}{4}$$      1 1 Domain: $$\left( {-\infty ,\infty } \right)$$   Range: $$\left( {-\infty ,\infty } \right)$$

$$f\left( {-x} \right)$$

Reflection

Flip graph around the $$y$$ axis

Every point on the graph is flipped around the $$y$$ axis.

The $$y$$’s stay the same; multiply the $$x$$ values by $$-1$$.

Parent:

$$y=\sqrt{x}$$

Transformed:

$$y=\sqrt{{-x}}$$

 –x       x y 0      0 0 –1      1 1 –4      4 2 Domain:  $$\left( {-\infty ,0} \right]$$     Range: $$\left[ {0,\,\,\infty } \right)$$

$$f\left( {\left| x \right|} \right)$$

Absolute Value on the $$x$$

“Throw away” the negative $$x$$’s; reflect the positive $$x$$’s across the $$y$$-axis.

The positive $$x$$’s stay the same; the negative $$x$$’s take on the $$y$$’s of the positive $$x$$’s.

Parent: $$y=\sqrt{x}$$

Transformed:

$$y=\sqrt{{\left| x \right|}}$$

 x y  New y –4 NA     2 –1 NA     1 0 0 1 1 4 2 Domain:  $$\left( {-\infty ,\infty } \right)$$     Range:  $$\left[ {0,\infty } \right)$$

Please visit Parent Functions and Transformations to see more!

Hope your math classes are going well this year  😆

Lisa

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