To model the data using an exponential function $f(x) = Ab^x$, follow these steps:

Step 1: Substitute $x = 0$ to find $A$.

$f(0) = A \cdot b^0 = A \cdot 1 = A$ From the table, $f(0) = 5$, so: $A = 5$

Step 2: Solve for $b$ using another point.

Using $x = 1$ and $f(1) = 2$: $f(1) = A \cdot b^1 = 5b$ $2 = 5b \implies b = \frac{2}{5} = 0.4$

Step 3: Verify the model with another point.

Using $x = 2$ and $f(2) = 0.8$: $f(2) = A \cdot b^2 = 5 \cdot (0.4)^2$ $f(2) = 5 \cdot 0.16 = 0.8$ This matches the given data, confirming the model is accurate.

Final Model:

$f(x) = 5 \cdot (0.4)^x$

Would you like a detailed explanation of any specific step or additional examples?

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Related Questions:

1. How do you find the base $b$ in an exponential model for more complex datasets?
2. How does changing $A$ affect the graph of an exponential function?
3. Can exponential decay be modeled when $b > 1$? Why or why not?
4. What is the difference between exponential growth and exponential decay?
5. How can logarithms be used to solve for $x$ in an exponential function?

Tip:

In exponential models, the base $b$ determines the rate of change: if $0 < b < 1$, the function represents decay; if $b > 1$, it represents growth.