Question

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

x y
−1
10
2 2160
(1 point)
The equation is $$
.

Answer 1

To find the exponential equation that fits the points given in the table, we can use the general form of an exponential function:

\[ y = ab^x \]

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

Given the points:

1. When \( x = -1 \), \( y = 10 \):
\[ 10 = ab^{-1} \]
or
\[ 10 = \frac{a}{b} \] (1)

2. When \( x = 2 \), \( y = 2160 \):
\[ 2160 = ab^2 \] (2)

Now we have two equations (1) and (2). We can express \( a \) in terms of \( b \) from equation (1):

From (1):

\[ a = 10b \]

Next, we can substitute \( a \) in equation (2):

\[ 2160 = (10b)b^2 \]

This simplifies to:

\[ 2160 = 10b^3 \]

Now, solve for \( b \):

\[ b^3 = \frac{2160}{10} \]
\[ b^3 = 216 \]
\[ b = \sqrt[3]{216} \]
\[ b = 6 \]

Now that we have \( b \), substitute it back into equation (1) to find \( a \):

\[ a = 10b \]
\[ a = 10 \times 6 = 60 \]

Now, we have both \( a \) and \( b \):

- \( a = 60 \)
- \( b = 6 \)

The exponential equation that contains the points is:

\[ y = 60(6^x) \]

Thus, the final answer is:

\[
y = 60 \cdot 6^x
\]