To solve this, we need to determine which of the given equations matches the exponential graph shown in the image. The options provided are:

A) $y = 2.4^x$
B) $y = 0.31^x$
C) $y = 0.45^x$
D) $y = 1.8^x$

Steps to Solve:

1. Identify Key Graph Characteristics:

   • The graph shows an exponential growth function because the curve rises steeply as $x$ increases.
   • Exponential growth functions have the general form $y = a^x$, where $a > 1$.

2. Eliminate Decreasing Functions:

   • $y = 0.31^x$ and $y = 0.45^x$ represent exponential decay (as $a < 1$, the graph would decrease as $x$ increases). These options can be eliminated.

3. Compare $y = 2.4^x$ and $y = 1.8^x$:

   • Both represent growth, so we need to test which one fits the graph.
   • From the graph, when $x = 1$, $y$ is approximately $2$. Testing both equations:
     • For $y = 2.4^x$: $y = 2.4^1 = 2.4$. This is too high compared to the graph.
     • For $y = 1.8^x$: $y = 1.8^1 = 1.8$. This is closer to the observed value of $2$.

4. Conclusion:
   The correct equation is D) $y = 1.8^x$.

Would you like detailed calculations for additional points or further clarification?

Related Questions:

1. What are the general rules to identify growth vs. decay in exponential functions?
2. How do you determine the base $a$ from a graph of an exponential function?
3. What happens to an exponential function as $x \to -\infty$?
4. Can exponential functions be used to model real-world phenomena? Give examples.
5. How does changing the base $a$ affect the steepness of an exponential function?

Tip:

When analyzing exponential graphs, test values like $x = 0$ and $x = 1$ for quick verification of the base $a$.