Question 6.10.12: Graphing an Exponential Function with a Base Between 0 and 1......
Graphing an Exponential Function with a Base Between 0 and 1
a) Graph y = (\frac{1}{2})^x.
b) Determine the domain and range of the function.
a) We begin by substituting values for x and calculating values for y. We then plot the ordered pairs and use these points to sketch the graph. To evaluate a fraction with a negative exponent, we use the fact that
\left ( \begin{matrix} \frac{a}{b} \end{matrix} \right )^{-x} = \left ( \begin{matrix} \frac{b}{a} \end{matrix} \right )^x
For example,
\left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^{-3} = \left ( \begin{matrix} \frac{2}{1} \end{matrix} \right )³ = 8
Then
y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^x
x = -3, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^{-3} = 2³ = 8
x = -2, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^{-2} = 2² = 4
x = -1, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^{-1} = 2¹ = 2
x = 0, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )^0 = 1
x = 1, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )¹ = \frac{1}{2}
x = 2, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )² = \frac{1}{4}
x = 3, y = \left ( \begin{matrix} \frac{1}{2} \end{matrix} \right )³ = \frac{1}{8}
The graph is illustrated in Fig. 6.50.
b) The domain is the set of all real numbers, \mathbb{R}. The range is y > 0.
| y | x |
| 8 | -3 |
| 4 | -2 |
| 2 | -1 |
| 1 | 0 |
| \frac{1}{2} | 1 |
| \frac{1}{4} | 2 |
| \frac{1}{8} | 3 |
