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We are given an exponential function \( f(x) = a \cdot b^x \) and a table of values to complete the equation.

The table shows:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 5 \\
1 & 8 \\
\hline
\end{array}
\]

### Step 1: Find \( a \)
From the first point, where \( x = 0 \) and \( f(x) = 5 \), we can substitute into the equation:
\[
f(0) = a \cdot b^0 = a \cdot 1 = a
\]
Thus, \( a = 5 \).

### Step 2: Find \( b \)
Now, using the second point where \( x = 1 \) and \( f(x) = 8 \), substitute into the equation:
\[
f(1) = 5 \cdot b^1 = 8
\]
This simplifies to:
\[
5b = 8
\]
Solving for \( b \):
\[
b = \frac{8}{5} = 1.6
\]

### Step 3: Write the complete equation
Now that we have \( a = 5 \) and \( b = 1.6 \), the equation for the function is:
\[
f(x) = 5 \cdot (1.6)^x
\]

Let me know if you need further details or clarifications.

Here are 5 related questions you can explore:
1. How does the growth factor \( b \) affect the behavior of an exponential function?
2. What would happen if \( b < 1 \) in the function \( f(x) = a \cdot b^x \)?
3. Can you find the value of \( f(x) \) when \( x = 2 \) using this equation?
4. How would the function change if the value of \( a \) was different?
5. What are the key characteristics of exponential functions in general?

**Tip**: To identify whether a function is exponential from a table, check if the ratio between consecutive outputs is constant when the inputs increase by 1.

Complete the exponential function f(x) based on the values in the table.

Learn how to solve the exponential function f(x) = a * b^x by using two points from a table of values. This problem involves solving for a and b based on the given data. Suitable for students in Grades 8-10.

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