Question

Use the table to answer the 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'll assume that the equation has the form:

\( y = ab^x \)

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

We have two points from the table:
1. \( (-1, 10) \)
2. \( (2, 2160) \)

Now we'll use these points to set up a system of equations.

### Step 1: Using the point \( (-1, 10) \)
Substituting into the equation:

\[
10 = ab^{-1}
\]
This can be rearranged to:
\[
10 = \frac{a}{b} \quad (1)
\]

### Step 2: Using the point \( (2, 2160) \)
Substituting into the equation:

\[
2160 = ab^2 \quad (2)
\]

### Step 3: Substitute Equation (1) into Equation (2)
We can express \( a \) in terms of \( b \) using Equation (1):

\[
a = 10b \quad (3)
\]

Now substitute Equation (3) into Equation (2):

\[
2160 = (10b)b^2
\]
\[
2160 = 10b^3
\]
\[
b^3 = \frac{2160}{10}
\]
\[
b^3 = 216
\]
\[
b = \sqrt[3]{216}
\]
\[
b = 6
\]

### Step 4: Finding \( a \)
Now substitute \( b \) back into Equation (3) to find \( a \):

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

### Step 5: Writing the equation
Now that we have \( a \) and \( b \):

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

Thus, the exponential equation that contains the points shown in the table is:

\[
\boxed{y = 60 \cdot 6^x}
\]