Question

make an exponential equation for this table
x y
1 12
2 36
3 108
4 324

give it in a form to write on a math paper

Answer 1

To find an exponential equation that fits the data given in the table, we can look for a general form of an exponential equation, which is:

\[ y = a \cdot b^x \]

where \( a \) and \( b \) are constants.

From the values in the table:

| x | y |
|---|----|
| 1 | 12 |
| 2 | 36 |
| 3 | 108 |
| 4 | 324 |

Let's determine the constants \( a \) and \( b \):

1. Start with the point when \( x = 1 \):
\[ y = a \cdot b^1 = 12 \]
Thus, we have:
\[ a \cdot b = 12 \quad (1) \]

2. For \( x = 2 \):
\[ y = a \cdot b^2 = 36 \]
Thus, we have:
\[ a \cdot b^2 = 36 \quad (2) \]

3. Dividing equation (2) by equation (1):
\[ \frac{a \cdot b^2}{a \cdot b} = \frac{36}{12} \]
\[ b = 3 \]

4. Now substitute \( b = 3 \) back into equation (1):
\[ a \cdot 3 = 12 \]
\[ a = 4 \]

Now we have both values:
- \( a = 4 \)
- \( b = 3 \)

Putting this together, the exponential equation is:

\[
\boxed{y = 4 \cdot 3^x}
\]

This is the exponential function that corresponds to the data points in your table.