Question

Find the equation of the exponential function represented by the table below:
xx yy
00 0, point, 0, 20.02
11 0, point, 0, 40.04
22 0, point, 0, 80.08
33 0, point, 1, 60.16

Answer 1

To find the equation of an exponential function represented by the given data points, we will denote the exponential function in the form:

\[ y = ab^x \]

where:
- \( a \) is the initial value (the value of \( y \) when \( x = 0 \)),
- \( b \) is the growth factor.

We will analyze the data points:

1. \( (0, 0.02) \)
2. \( (1, 0.04) \)
3. \( (2, 0.08) \)
4. \( (3, 0.16) \)

From the first data point, we can immediately determine that:

\[
a = 0.02
\]

Now we can apply the other data points to find \( b \).

Using the point \( (1, 0.04) \):

\[
0.04 = 0.02b^1
\]

Dividing both sides by \( 0.02 \):

\[
b = \frac{0.04}{0.02} = 2
\]

Now, we will verify \( b \) with the other data points.

Using the point \( (2, 0.08) \):

\[
0.08 = 0.02b^2
\]

Substituting \( b = 2 \):

\[
0.08 = 0.02(2^2) = 0.02 \cdot 4
\]

Calculating:

\[
0.02 \cdot 4 = 0.08 \quad \text{(TRUE)}
\]

Now check the point \( (3, 0.16) \):

\[
0.16 = 0.02b^3
\]

Substituting again \( b = 2 \):

\[
0.16 = 0.02(2^3) = 0.02 \cdot 8
\]

Calculating:

\[
0.02 \cdot 8 = 0.16 \quad \text{(TRUE)}
\]

All points are consistent. Thus, the parameters we found are correct.

The equation of the exponential function is:

\[
y = 0.02 \cdot 2^x
\]