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Given the graph of the exponential equation, write the correct exponential equation for the graph. (1 point) Responses y=32(3)x y equals Start Fraction 3 over 2 End Fraction left parenthesis 3 right parenthesis superscript x baseline y=3(23)x y equals 3 left parenthesis Start Fraction 2 over 3 End Fraction right parenthesis superscript x baseline y=2(32)x y equals 2 left parenthesis Start Fraction 3 over 2 End Fraction right parenthesis superscript x baseline y=3(32)x

We can determine the correct exponential equation for this graph by analyzing the points provided.

### Steps:
1. **Identify key points on the graph:**
   - The graph passes through the points \((0, 3)\) and \((1, 4.5)\).

2. **General form of the exponential equation:**
   The equation of an exponential function is generally of the form:
   \[
   y = a \cdot b^x
   \]
   Where:
   - \(a\) is the initial value (when \(x = 0\)),
   - \(b\) is the base of the exponential function.

3. **Find the value of \(a\):**
   From the point \((0, 3)\), when \(x = 0\), \(y = a\). Hence:
   \[
   a = 3
   \]
   So the equation becomes:
   \[
   y = 3 \cdot b^x
   \]

4. **Determine the base \(b\):**
   Using the second point \((1, 4.5)\), substitute \(x = 1\) and \(y = 4.5\) into the equation:
   \[
   4.5 = 3 \cdot b^1
   \]
   Solving for \(b\):
   \[
   b = \frac{4.5}{3} = 1.5
   \]

5. **Write the final equation:**
   The exponential equation is:
   \[
   y = 3 \cdot (1.5)^x
   \]

### Verifying with the given options:
- The correct option matches **\(y = 3 \left(\frac{3}{2}\right)^x\)**.

### Final Answer:
The correct equation is:
\[
y = 3 \left(\frac{3}{2}\right)^x
\]

Do you need any further clarifications or details?

### Related Questions:
1. How can you verify whether a graph is exponential just by looking at it?
2. What is the significance of the base in an exponential function?
3. How would the graph change if the base \(b\) were less than 1?
4. What is the effect of changing the initial value \(a\) on the graph?
5. Can an exponential function have a negative base? Why or why not?

**Tip:** Always check key points on the graph to identify important values like the initial value and the base of the exponential function.

Determine the correct exponential equation for a graph passing through points (0,3) and (1,4.5). Learn how to calculate the initial value and the base to form the exponential equation y = 3(3/2)^x. Suitable for Grades 9-12.

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