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 in the table, we can use the general form of an exponential function:

\[
y = ab^x
\]

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

The points given in the table are \((-1, 10)\) and \( (2, 2160)\).

### Step 1: Use the first point to set up the equation

Using the point \((-1, 10)\):

\[
10 = ab^{-1}
\]

This simplifies to:

\[
10 = \frac{a}{b} \tag{1}
\]

### Step 2: Use the second point to set up another equation

Using the point \((2, 2160)\):

\[
2160 = ab^{2}
\]
\tag{2}

### Step 3: Solve the equations

From equation (1):
\[
a = 10b \tag{3}
\]

Now, substitute equation (3) into equation (2):

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

This simplifies to:

\[
2160 = 10b^{3}
\]

Now divide both sides by 10:

\[
216 = b^{3}
\]

Now take the cube root of both sides:

\[
b = \sqrt[3]{216} = 6
\]

### Step 4: Find \( a \)

Now, substitute \( b = 6 \) back into equation (3) to find \( a \):

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

### Step 5: Write the final exponential equation

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

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

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

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