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The table provided suggests an exponential function of the form:

\[
y = a \cdot b^x
\]

where \(a\) is the initial value (when \(x = 0\)), and \(b\) is the growth/decay factor. Let's analyze the table:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & 40 \\
0 & 20 \\
1 & 10 \\
2 & 5 \\
\hline
\end{array}
\]

1. When \(x = 0\), \(y = 20\), so \(a = 20\).
2. To find \(b\), note how the \(y\)-values change as \(x\) increases. Each time \(x\) increases by 1, \(y\) is halved (from 40 to 20, 20 to 10, and 10 to 5). This indicates that \(b = \frac{1}{2}\).

Thus, the equation for the exponential function is:

\[
y = 20 \cdot \left(\frac{1}{2}\right)^x
\]

Would you like more details or clarifications?

Here are 5 related questions:
1. How do we identify the base \(b\) in exponential functions from a table of values?
2. What happens if the base \(b\) is greater than 1?
3. How can we solve for \(x\) when given a specific \(y\)-value in an exponential equation?
4. How do exponential growth and decay differ in form and behavior?
5. Can we find the inverse of an exponential function like the one we derived?

**Tip:** Always check if the ratio between consecutive \(y\)-values is consistent; this helps confirm the exponential nature of the function!

The table of ordered pairs (x, y) gives an exponential function. Write an equation for the function. The table is as follows: 

 x | y
-1 | 40
 0 | 20
 1 | 10
 2 | 5

Learn how to find the exponential function equation using a table of values. This problem involves understanding exponential decay, the function y = a * b^x, and recognizing patterns in sequences. Suitable for students in Grades 8-10.

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