On a coordinate plane, an exponential function that has a horizontal asymptote of y = 0 is shown. The function has points (negative 2, StartFraction 4 Over 81 EndFraction), (negative 1, two-ninths), (0, 1), (1, nine-halves), (2, StartFraction 81 Over 4 EndFraction)
What is the multiplicative rate of change of the exponential function shown on the graph?

Two-ninths
1
4
Nine-halves

We are given an exponential function that passes through the points \((-2, \frac{4}{81}), (-1, \frac{2}{9}), (0, 1), (1, \frac{9}{2}), (2, \frac{81}{4})\) and has a horizontal asymptote of \(y = 0\). To determine the multiplicative rate of change of this exponential function, we need to observe how the function values change as the \(x\)-values increase by 1 unit.

### Step-by-Step Approach:

An exponential function is generally of the form:

\[
y = ab^x
\]

where:
- \(a\) is the initial value (value of \(y\) when \(x = 0\)),
- \(b\) is the base, which represents the **multiplicative rate of change**.

Looking at the given point \((0, 1)\), we know that when \(x = 0\), \(y = 1\). So, \(a = 1\). This simplifies the equation to:

\[
y = b^x
\]

Now, let's use another point to determine \(b\). Take the point \((1, \frac{9}{2})\). Plugging this into the equation \(y = b^x\):

\[
\frac{9}{2} = b^1
\]

This gives us \(b = \frac{9}{2}\).

Thus, the multiplicative rate of change of the exponential function is:

\[
\boxed{\frac{9}{2}}
\]

### Conclusion:
The multiplicative rate of change of the exponential function is \(\frac{9}{2}\), which corresponds to the answer **Nine-halves**.

Would you like further details or have any questions?

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Here are 5 related questions to explore:
1. What is the general form of an exponential function?
2. How does the base \(b\) in an exponential function affect its growth or decay?
3. Can an exponential function have a negative base? Why or why not?
4. How would you graph an exponential function using a table of values?
5. What is the significance of the horizontal asymptote in an exponential function?

**Tip**: In exponential functions, the base \(b\) determines whether the function represents growth (\(b > 1\)) or decay (\(0 < b < 1\)).

On a coordinate plane, an exponential function that has a horizontal asymptote of y = 0 is shown. The function has points (-2, 4/81), (-1, 2/9), (0, 1), (1, 9/2), (2, 81/4). What is the multiplicative rate of change of the exponential function shown on the graph? Two-ninths, 1, 4, Nine-halves.

Learn how to determine the multiplicative rate of change for an exponential function with a horizontal asymptote of y = 0. This problem involves points (-2, 4/81), (-1, 2/9), (0, 1), (1, 9/2), (2, 81/4) and uses the formula y = b^x to solve for the base b. Suitable for students in Grades 9-12.

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