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Wize High School Grade 12 Pre-Calculus Textbook > Exponential Functions

Transformations of Exponential Functions

Characteristics of Exponential FunctionsPrevious Section
Applications of Exponential FunctionsNext Section
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Transformations of Exponential Functions

Let y=a(b)c(x−h)+ky=a(b)^{c(x-h)}+k be the transformed function of y=f(x), y=f(x),~ where a, b, c, h, k are real numbers. Then:

HorizontalVerticalc<0‾:a<0‾:horizontal reflection about thevertical reflection about they-axisx-axis∣c∣>1‾:∣a∣>1‾:horizontal compression of  1c  unitsvertical expansion of ’a’ units∣c∣<1‾:∣a∣<1‾:horizontal expansion of  1c  unitsvertical compression of ’a’ unitsh>0‾:k>0‾:horizontal translation ’h’ units rightvertical translation ’k units uph<0‾:k<0‾:horizontal translation ’h’ units leftvertical translation ’k’ units down\begin{array}{l c c l} \text{Horizontal}&&&\text{Vertical}\\\\ \underline{c<0}:&&&\underline{a<0}:\\ \text{horizontal reflection about the}&&&\text{vertical reflection about the}\\ \text{y-axis}&&&\text{x-axis}\\\\\\ \underline{|c|>1}:&&&\underline{|a|>1}:\\ \text{horizontal compression of}~~\displaystyle\frac{1}{c}~~\text{units}&&&\text{vertical expansion of 'a' units}\\\\\\ \underline{|c|<1}:&&&\underline{|a|<1}:\\ \text{horizontal expansion of}~~\displaystyle\frac{1}{c}~~\text{units}&&&\text{vertical compression of 'a' units}\\\\\\ \underline{h>0}:&&&\underline{k>0}:\\ \text{horizontal translation 'h' units right}&&&\text{vertical translation 'k units up}\\\\\\ \underline{h<0}:&&&\underline{k<0}:\\ \text{horizontal translation 'h' units left}&&&\text{vertical translation 'k' units down} \end{array}

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Example

The graph of y=−2(2)12(x−1)+1y=-2(2)^{\frac{1}{2}(x-1)}+1 has the following transformations:
  • Horizontal:
  • Expansion by a factor of 2
  • Translation 1 unit right
  • Vertical:
  • Reflection about the x-axis
  • Expansion by a factor of 2
  • Translation 1 unit up

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Parent Function: y=2xy=2^x
Table of values for parent function:

xy−21/4−11/2011224\begin{array}{|c|c|}\hline x&y\\\hline -2&1/4\\\hline -1&1/2\\\hline 0&1\\\hline 1&2\\\hline 2&4\\\hline \end{array}

Transformed Function: y=−2(2)12(x−1)+1y=-2(2)^{\frac{1}{2}(x-1)}+1
Table of values for transformed function:

xy−31/2−101−13−35−9\begin{array}{|c|c|}\hline x&y\\\hline -3&1/2\\\hline -1&0\\\hline 1&-1\\\hline 3&-3\\\hline 5&-9\\\hline \end{array}


The horizontal asymptote is y=1y=1
The domain is x∈Rx\in\mathbb{R}
The range is y<1y<1
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Example: Transformations of Exponential Functions

Sketch a graph of the function y=2(13)2−x−1y=2\Bigg(\dfrac{1}{3}\Bigg)^{2-x}-1, stating the domain, range, and horizontal asymptote.


First, let's write the base of the function yy as a whole number, not a rational number, using the rule 1an=a−n\colorFive{\dfrac{1}{a^n}=a^{-n}}.
y=2(13)2−x−1=2(3−1)2−x−1=2(3)x−2−1\begin{array}{rcl} y&=&2\Bigg(\dfrac{1}{3}\Bigg)^{2-x}-1\\\\ &=&2(3^{-1})^{2-x}-1\\\\ &=&2(3)^{x-2}-1 \end{array}

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Therefore, the parent function is y=3xy=3^x.
  • The domain isx∈Rx\in\mathbb{R}
  • The range is y>0y>0
  • The horizontal asymptote is y=0y=0
The table of values for y=3xy=3^x:

xy−21/9−11/3011329\begin{array}{c|c} x&y\\\hline -2&1/9\\\\ -1&1/3\\\\ 0&1\\\\ 1&3\\\\ 2&9 \end{array}


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The transformed function is y=2(3)x−2−1y=2(3)^{x-2}-1.
  • Horizontal:
  • Translation 2 right
  • Vertical
  • Expansion by a factor of 2
  • Translation 1 unit down

The table of values and graph for y=2(3)x−2−1y=2(3)^{x-2}-1:

xy0−7/91−1/32135417\begin{array}{c|c} x&y\\\hline 0&-7/9\\\\ 1&-1/3\\\\ 2&1\\\\ 3&5\\\\ 4&17 \end{array}

  • The domain is x∈Rx\in\mathbb{R}
  • The range is y>−1y>-1
  • The horizontal asymptote is y=−1y=-1

Practice: Transformations of Exponential Functions

Which of the following is a graph of y=−12(2)xy=-\dfrac{1}{2}(2)^{x}.

Practice: Transformations of Exponential Functions

The point (2,16)(2,16) is on the graph y=bxy=b^x. What point must be on the graph of y=2(1b)x−2−1y=2\Bigg(\dfrac{1}{b}\Bigg)^{x-2}-1?

Practice: Transformations of Exponential Functions

Sketch a graph of y=12(15)1−x+2y=\dfrac{1}{2}\Bigg(\dfrac{1}{5}\Bigg)^{1-x}+2. Identify the domain, range, and horizontal asymptote.
Extra Practice
Exponential Functions: Transformations
Match the function to the correct graph.
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
Sketch a graph of y=−(32)x+3y=-\Bigg(\dfrac{3}{2}\Bigg)^{x}+3. Identify the domain, range, and horizontal asymptote.
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
Sketch a graph of y=53(12)x−3−1y=\dfrac{5}{3}\Bigg(\dfrac{1}{2}\Bigg)^{x-3}-1. Identify the domain, range, and horizontal asymptote.
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
The point (5,−8)(5,-8) is on the graph y=bxy=b^x. What point must be on the graph of y=32(b)4x+6y=\dfrac{3}{2}(b)^{4x+6}?
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
The point (−3,4)(-3,4) is on the graph y=bxy=b^x. What point must be on the graph of y=−3(1b)2x+4+2y=-3\Bigg(\dfrac{1}{b}\Bigg)^{2x+4}+2?
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
Which of the following is a graph of y=−2(12)xy=-2\Bigg(\dfrac{1}{2}\Bigg)^{x}.
Transformations of Exponential Functions
Practice: Transformations of Exponential Functions
Which of the following is a graph of y=23(3)xy=\dfrac{2}{3}(3)^{x}.