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To determine the equation of the exponential function from the graph, we observe the following points:

1. At \( x = 0 \), \( y = 2 \). This indicates that the initial value (or the coefficient) is 2.
2. At \( x = 1 \), \( y = 6 \). To find the base \( b \) of the function \( y = ab^x \), where \( a = 2 \), we use this point.

Substitute \( x = 1 \) and \( y = 6 \):
\[
6 = 2b^1
\]
Solving for \( b \), we get:
\[
b = \frac{6}{2} = 3
\]

Thus, the equation of the function is:
\[
y = 2(3)^x
\]

The correct answer is:
**\( y = 2(3)^x \)**.

Would you like more details or have any questions about this? Here are some related questions for further practice:

1. How do you determine the base of an exponential function given two points on its graph?
2. What changes in the graph when the base of an exponential function is less than 1?
3. How does changing the initial value \( a \) affect the shape of the exponential graph?
4. Can an exponential function have a negative base? Why or why not?
5. How do you distinguish between exponential growth and decay functions?

**Tip:** In an exponential function, the initial value is found where \( x = 0 \), and the base represents the factor by which the function grows or shrinks for each unit increase in \( x \).

The graph shows an exponential function. What is the equation of the function?

Learn how to determine the equation of an exponential function from a graph. This example uses points (0, 2) and (1, 6) to find the equation y = 2(3)^x, covering key concepts like initial value and base calculation. Suitable for students in Grades 9-12.

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