Question

This table shows some values of an exponential function, h(x). Complete the table.
Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
x h(x)
–
4 486
–
3 162
–
2
–
1
0

Answer 1

To complete the table for the exponential function \( h(x) \), we can start by observing the values provided for \( h(x) \) at \( x = -4 \) and \( x = -3 \).

Given:
- \( h(-4) = 486 \)
- \( h(-3) = 162 \)

An exponential function typically has the form \( h(x) = a \cdot b^x \). We can use the values we have to find constants \( a \) and \( b \).

First, we calculate the ratio of \( h(-3) \) to \( h(-4) \):

\[
\frac{h(-3)}{h(-4)} = \frac{162}{486} = \frac{1}{3}
\]

Since the function is exponential, this ratio can be expressed as follows:

\[
\frac{h(-3)}{h(-4)} = \frac{b^{-3}}{b^{-4}} = b^{-3 + 4} = b^1 = b
\]

Thus, we find that \( b = \frac{1}{3} \).

Next, we can use one of the known values to solve for \( a \). We use \( h(-4) \):

\[
h(-4) = a \cdot b^{-4} = a \cdot \left(\frac{1}{3}\right)^{-4} = a \cdot 81
\]

Setting this equal to 486:

\[
a \cdot 81 = 486
\]

Solving for \( a \):

\[
a = \frac{486}{81} = 6
\]

Now we have the function:

\[
h(x) = 6 \cdot \left(\frac{1}{3}\right)^x
\]

Next, we can calculate \( h(x) \) for the missing values of \( x \):

1. **For \( x = -2 \)**:
\[
h(-2) = 6 \cdot \left(\frac{1}{3}\right)^{-2} = 6 \cdot 9 = 54
\]

2. **For \( x = -1 \)**:
\[
h(-1) = 6 \cdot \left(\frac{1}{3}\right)^{-1} = 6 \cdot 3 = 18
\]

3. **For \( x = 0 \)**:
\[
h(0) = 6 \cdot \left(\frac{1}{3}\right)^{0} = 6 \cdot 1 = 6
\]

Now we can fill in the completed table:

| x | h(x) |
|-----|------|
| -4 | 486 |
| -3 | 162 |
| -2 | 54 |
| -1 | 18 |
| 0 | 6 |

So the completed values for \( h(x) \) are:
- \( h(-2) = 54 \)
- \( h(-1) = 18 \)
- \( h(0) = 6 \)