To model the data using an exponential function $f(x) = Ab^x$, we need to determine the constants $A$ and $b$ based on the values provided:

Given dataTo model the data using an exponential function $f(x) = Ab^x$, we need to determine the constants $A$ and $b$ based on the values provided:

Given

• When $x = 0$, $f(x) = 50$.
• When $x = 1$, $f(x) = 30$.
• When $x = 2$, $f(x) = 18$.

Step 1: Determine $A$

Since $f(0) = 50$, and substituting $x = 0$ in $f(x) = Ab^x$, we get: $f(0) = A \cdot b^0 = A = 50$ So, $A = 50$.

Step 2: Solve for $b$

Now that $A = 50$, we substitute this into the equations for $f(1)$ and $f(2)$ to find $b$.

1. For $f(1) = 30$: $50 \cdot b^1 = 30$ $b = \frac{30}{50} = 0.6$

2. Verifying with $f(2) = 18$: $50 \cdot (0.6)^2 = 50 \cdot 0.36 = 18$ This matches the given value, confirming that $b = 0.6$ is correct.

Final Model

The exponential function that models the data is: $f(x) = 50 \cdot (0.6)^x$

Would you like further details on the steps, or have any other questions?

---

Here are five related questions for further exploration:

1. How would the model change if the values of $f(x)$ were different?
2. What does the constant $A$ represent in an exponential model?
3. How can we interpret the decay factor $b$ in the context of this function?
4. How would we use this function to predict $f(x)$ for values of $x$ not in the table?
5. How would the model differ if we had an exponential growth rather than decay?

Tip: Exponential models are useful for data showing rapid growth or decay and can be transformed to linear models by taking logarithms, simplifying calculations in some cases.