• If \( {\color{BrickRed}a} \lt 0\), then \( y = c^x \) is reflected in the \( x \)-axis.
• \( y = c^x \) is stretched vertically about the \( x \)-axis by a factor of \( \lvert {\color{BrickRed}a} \rvert \).
• If \( {\color{NavyBlue}b} \lt 0\), then \( y = c^x \) is reflected in the \( y \)-axis.
• \( y = c^x \) is stretched horizontally about the \( y \)-axis by a factor of \( \dfrac{1}{\lvert {\color{NavyBlue} b} \rvert} \).
• \( y = c^x \) is translated horizontally \( \color{Mulberry} h \) units.

If \({\color{Mulberry}h} \gt 0\), then \(y=c^x\) is translated right.

If \({\color{Mulberry}h} \lt 0\), then \(y=c^x\) is translated left.

• \( y = c^x \) is translated vertically \( \color{Violet}k \) units.

If \({\color{Violet}k} \gt 0\), then \(y=c^x\) is translated up.

If \({\color{Violet}k} \lt 0\), then \(y=c^x\) is translated down.

The transformation of each point is defined by the mapping

When applying the transformations to the graph of the function, the stretches and/or reflections must be performed first (in any order) prior to the translations.

Sketch the graph of \( f(x) = 3^x \). Describe the transformations applied to \( f(x) \) to obtain the graph of the function \( g(x)= \textcolor{BrickRed}{-\dfrac{1}{2}} \left (3^{x-\textcolor{Mulberry}{2}} \right)+\textcolor{Violet}{4} \) and graph the transformed function.

Solution

Using the points \(\left (-1, \frac{1}{3}\right ), (0,1),(1,3), \) and \( (2,9) \), we sketch the graph of \( f(x)=3^x \).
Identify the transformations applied to \(f(x)\) to obtain \(g(x)\).

The graph of \( g(x)=-\dfrac{1}{2}\left (3^{x-2} \right )+4 \) is obtained by applying the transformations, in order:

• a reflection in the \(x\)-axis \(\left ( a = -\frac{1}{2} \lt 0\right) \)
• a vertical stretch about the \(x\)-axis by a factor of \( \frac{1}{2} \) \( \left(\lvert a \rvert = \frac{1}{2}\right) \)
• a horizontal translation right \(2\) units \( ( h = 2) \)
• a vertical translation up \(4\) units \( (k = 4) \)

Sketch the graph of \( f(x) = 3^x \). Describe the transformations applied to \( f(x) \) to obtain the graph of the function \( g(x)= \textcolor{BrickRed}{-\dfrac{1}{2}} \left (3^{x-\textcolor{Mulberry}{2}} \right)+\textcolor{Violet}{4} \) and graph the transformed function.

Mapping Notation

The points on the new function can be obtained from the mapping: \( (x, y) \rightarrow \left( x + {\color{Mulberry}2}, {\color{BrickRed}-\dfrac{1}{2}}y + {\color{Violet}4} \right) \).

Specifically,

Thus, \( g(x) \) is a decreasing function with a horizontal asymptote of \( y = 4 \).

The domain is \( \{ x \mid x \in \mathbb{R} \} \). The range is \( \{ y \mid y \lt 4,\ y \in \mathbb{R} \} \).