To determine which equation represents the graph, let's carefully analyze it:

Observations from the graph:

1. Shape: The graph is exponential and passes through the y-axis at (0, 2), which indicates the initial value $y = 2$ when $x = 0$.
2. Growth: The graph increases exponentially as $x$ increases, suggesting a positive base greater than 1.

Step-by-step check of the options:

1. Option 1: $y = 2(3)^x$

   • This represents exponential growth with an initial value $2$ and a growth factor of $3$.
   • At $x = 0$, $y = 2(3)^0 = 2$ ✅
   • As $x$ increases, $y$ grows quickly (matching the observed graph).

2. Option 2: $y = 3(2)^x$

   • Here, the initial value at $x = 0$ is $y = 3$, which does not match the graph $y = 2$. ❌

3. Option 3: $y = 2\left(\frac{1}{3}\right)^x$

   • This represents exponential decay, as the base $\frac{1}{3}$ is less than 1.
   • The graph is growing, not decaying. ❌

4. Option 4: $y = (2)^x + 3$

   • This equation has a base of $2$, but the graph would shift upward by 3 units.
   • At $x = 0$, $y = 2^0 + 3 = 4$, which does not match $y = 2$. ❌

Correct Answer:

The correct equation is $y = 2(3)^x$.

Would you like a deeper explanation of exponential functions? Let me know!

Related Questions:

1. What is the general form of an exponential function?
2. How do you determine the growth factor from a graph?
3. What happens to an exponential function as $x \to -\infty$?
4. How do you graph $y = a(b)^x$ step by step?
5. How does an exponential decay graph differ from an exponential growth graph?

Tip:

The base $b$ in $y = a(b)^x$ determines if the function grows ($b > 1$) or decays ($0 < b < 1$).