To write an exponential equation that contains the points (0, 2) and (3, 54), we need to find the general form of an exponential function, which is given by:
\[ y = a \cdot b^x \]
Where:
- \( a \) is the initial value when \( x = 0 \)
- \( b \) is the base of the exponential function
- \( x \) is the input value (in this case, the x-coordinate of the points)
Given points:
- (0, 2): When \( x = 0 \), \( y = 2 \)
- (3, 54): When \( x = 3 \), \( y = 54 \)
1. Substitute the values of the first point (0, 2) into the exponential equation:
\[ 2 = a \cdot b^0 \]
\[ 2 = a \]
2. Now, we have the value of \( a \), which is 2. Next, substitute the values of the second point (3, 54) into the equation:
\[ 54 = 2 \cdot b^3 \]
3. Simplify the equation:
\[ 54 = 2 \cdot b^3 \]
\[ 27 = b^3 \]
\[ b = 3 \]
4. Therefore, the exponential equation that passes through the points (0, 2) and (3, 54) is:
\[ y = 2 \cdot 3^x \]
This equation represents the exponential relationship between x and y based on the given points.
