Question

Write the exponential equation that contains the points shown in the table.

x y
−1
200
2 0.675
(1 point)

Answer 1

To find an exponential equation of the form \( y = ab^x \) that passes through the points \((-1, 200)\) and \((2, 0.675)\), we can set up a system of equations based on these points.

1. From the point \((-1, 200)\):
\[
200 = ab^{-1} \quad \text{(1)}
\]
This simplifies to:
\[
200 = \frac{a}{b} \quad \text{(2)}
\]

2. From the point \((2, 0.675)\):
\[
0.675 = ab^2 \quad \text{(3)}
\]

Now we have two equations:
1. \(200 = \frac{a}{b}\)
2. \(0.675 = ab^2\)

From equation (2), we can express \(a\) in terms of \(b\):
\[
a = 200b \quad \text{(4)}
\]

Now we substitute equation (4) into equation (3):
\[
0.675 = (200b)b^2
\]
\[
0.675 = 200b^3
\]
\[
b^3 = \frac{0.675}{200}
\]
\[
b^3 = 0.003375
\]

Now we take the cube root of both sides to solve for \(b\):
\[
b = \sqrt[3]{0.003375} = 0.15
\]

Now, substitute \(b\) back into equation (4) to find \(a\):
\[
a = 200 \times 0.15 = 30
\]

Thus, we have \(a = 30\) and \(b = 0.15\). Therefore, the exponential equation is:
\[
y = 30(0.15)^x
\]

This is the final result for the exponential equation that contains the points in the table.