Question

Find the equation of the exponential function represented by the table below:
xx yy
00 55
11 2020
22 8080
33 320320

Answer 1

To find the equation of the exponential function based on the given data points, we first need to examine the values in the table:

```
x | y
---------
0 | 5
1 | 20
2 | 80
3 | 320
```

Looking at the \( y \) values, we can observe a pattern. We will try to express \( y \) as a function of \( x \) in the form:

\[
y = ab^x
\]

where \( a \) is the value of the function when \( x = 0 \) and \( b \) is the base of the exponential function.

From the first data point at \( x = 0 \):

\[
y(0) = a = 5
\]

Thus, we have:

\[
y = 5b^x
\]

Next, we can use any of the other points to find \( b \). Let's use the point \( (1, 20) \):

\[
y(1) = 5b^1 = 20
\]

Solving for \( b \):

\[
5b = 20
\]
\[
b = \frac{20}{5} = 4
\]

Now we have both \( a \) and \( b \) for the function:

\[
y = 5 \cdot 4^x
\]

We should verify this equation with the other points in the table:

1. For \( x = 2 \):

\[
y = 5 \cdot 4^2 = 5 \cdot 16 = 80
\]

2. For \( x = 3 \):

\[
y = 5 \cdot 4^3 = 5 \cdot 64 = 320
\]

All points are consistent with the derived equation. Therefore, the equation of the exponential function that fits the given data is:

\[
\boxed{y = 5 \cdot 4^x}
\]