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:

\(\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

\(y=\sqrt[3]{x}\)
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.

<
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:  \(y=\sqrt[3]{x}\)

 

Transformed: \(y=\left| {\sqrt[3]{x}} \right|\)

 

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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