An exponential function of the form \( y = c^x, \ c \gt 0, \ c \neq 1 \), has

• a domain: \( \{ x \mid x \in \mathbb{R} \} \)
• a range: \( \{ y \mid y \gt 0, y \in \mathbb{R} \} \)
• no \( x \)-intercept and a \( y \)-intercept at \( (0, 1) \), and
• a horizontal asymptote of \( y = 0 \) (the \( x \)-axis).

The graph of an exponential function, \(y=c^x\) where \( c \gt 1 \), is always increasing. The greater the value of the base, \( c \), the faster the curve increases.

The graph of an exponential function, \(y=c^x\) where \( 0 \lt c \lt 1 \), is always decreasing. The smaller the value of the base, \( c \), the faster the curve decreases.

Consider the table of values and graphs of \( y = 2^x \) and \( y= \left( \frac{1}{2} \right)^x \).

The graph of \( y = \left( \frac{1}{2} \right)^x \) is a reflection of the graph \( y = 2^x \) in the \( y \)-axis.

This observation is supported by the fact that \( y = \left( \frac{1}{2} \right)^x \) can be expressed as \( y = 2^{-x} \).

Consider the table of values and graphs of \( y = 2^x \) and \( y= \left( \frac{1}{2} \right)^x \).

Why must \( c \gt 0\) and \( c \neq 1 \)?

When \(c = 1 \), the graph is the horizontal line \( y = 1 \). That is,

\( y = 1^x = 1\), for all \(x \in \mathbb{R} \)

Similarly, when \( c = 0\), \(y=0^x\).

Thus \(y=0\) when \(x\gt 0\), and \(y\) will be undefined when \(x\lt 0\) as this implies division by \(0\).

When \(x=0\), \(y=0^0\), which some consider to be undefined.

For our purposes, we will follow the convention that

What happens when \( c \lt 0 \)?

For example, consider \( y = (-2)^x \):

• When \( x \) is even, \( y \) is positive; e.g., \( (-2)^2 = 4, (-2)^4 = 16, \dots \)
• When \( x \) is odd, \( y \) is negative; e.g., \( (-2)^3 =-8, (-2)^5 = -32, \dots \)
• When \( x \) is \( \frac{1}{2} \) or \( \frac{1}{4} \), \( y \) is not a real number; e.g., \( y = \sqrt{-2}, y = \sqrt[4]{-2} \).

This relation lacks continuity and the previously discussed properties of the exponential functions are lost.

We will now apply our prior knowledge of transformations to sketch the graphs of exponential functions of the form

Given

can you recall the role of the parameters \( {\color{BrickRed}a}, {\color{NavyBlue}b}, {\color{Mulberry}h}, \) and \( {\color{Violet}k} \) in transforming the function \( y = f(x) \)?