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.