To find an exponential equation that fits the given data, we can first observe the relationship between the values of \( x \) and \( y \). The given points are:
x: 1, 2, 3, 4
y: 12, 36, 108, 324
We can see if the relationship between \( x \) and \( y \) can be described by an equation of the form:
\[ y = a \cdot b^x \]
where \( a \) is a constant and \( b \) is the base of the exponential function.
- Start by finding the ratio of successive \( y \) values:
- \( \frac{y_2}{y_1} = \frac{36}{12} = 3 \)
- \( \frac{y_3}{y_2} = \frac{108}{36} = 3 \)
- \( \frac{y_4}{y_3} = \frac{324}{108} = 3 \)
From the calculations above, we see that the ratios are constant, indicating that \( b = 3 \).
- Now we need to find the value of \( a \). Let’s use one of the points; when \( x = 1 \) and \( y = 12 \):
\[ y = a \cdot b^x \implies 12 = a \cdot 3^1 \]
This simplifies to:
\[ 12 = 3a \implies a = \frac{12}{3} = 4 \]
- Now we can write the exponential equation:
\[ y = 4 \cdot 3^x \]
We can verify this equation by plugging in the other values of \( x \):
-
For \( x = 2 \): \[ y = 4 \cdot 3^2 = 4 \cdot 9 = 36 \]
-
For \( x = 3 \): \[ y = 4 \cdot 3^3 = 4 \cdot 27 = 108 \]
-
For \( x = 4 \): \[ y = 4 \cdot 3^4 = 4 \cdot 81 = 324 \]
All values match the original table. Thus, the exponential equation that fits the data is:
\[ \boxed{y = 4 \cdot 3^x} \]
1 answer