Question

Write an exponential equation that contains the points (-1, 10) and (2, 2160)

Answer 1

To find an exponential equation of the form \( y = ab^x \) that passes through the points (-1, 10) and (2, 2160), we need to set up a system of equations based on these points.

Starting with the general form:

1. For the point (-1, 10):
\[
10 = ab^{-1} \quad \text{(1)}
\]
This simplifies to:
\[
10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b \quad \text{(2)}
\]

2. For the point (2, 2160):
\[
2160 = ab^2 \quad \text{(3)}
\]

Now we can substitute equation (2) into equation (3):
\[
2160 = (10b)b^2
\]
\[
2160 = 10b^3
\]
\[
b^3 = \frac{2160}{10}
\]
\[
b^3 = 216
\]
\[
b = \sqrt[3]{216}
\]
\[
b = 6
\]

Now that we have found \( b = 6 \), we can substitute it back into equation (2) to find \( a \):
\[
a = 10b = 10 \cdot 6 = 60
\]

So, we have \( a = 60 \) and \( b = 6 \). Therefore, the exponential equation is:
\[
y = 60 \cdot 6^x
\]

This is the exponential equation that contains the points (-1, 10) and (2, 2160).