The problem asks for the equation that matches the graph shown. The graph appears to resemble an exponential function, where the curve grows rapidly as $x$ increases.

Observing the Graph:

1. The graph passes through the point (0, -2), suggesting that the equation likely has a constant shift (possibly a -2).
2. It also appears to be an exponential growth, based on the steep upward increase as $x$ increases, suggesting the presence of an exponential term like $3^x$.

Review of the Equation Options:

Let's analyze each of the options:

• Option A: $y = 2 \cdot 3^{x-1} - 2$

  • This is an exponential function, shifted horizontally by 1 unit (due to $x - 1$) and vertically by -2 units. This looks promising based on the shape of the graph and the y-intercept at -2.

• Option B: $y = (-3) \cdot 2^{x-1} - 2$

  • This equation has a base of 2, and a negative coefficient (-3). This would result in a decreasing function, not matching the graph.

• Option C: $y = 3 \cdot 2^{x+1} + 2$

  • This equation also has a base of 2 and suggests a positive vertical shift, which does not match the graph's y-intercept at -2.

• Option D: $y = 2 \cdot 3^{x-1} - 2$

  • This is the same as Option A, so it looks correct.

• Option E: $y = (-2) \cdot 3^{1-x} + 2$

  • The negative sign would result in a decreasing function, which does not match the graph.

Conclusion:

The correct equation that corresponds to the graph is Option A (or Option D, since they are the same):
$y = 2 \cdot 3^{x-1} - 2$

Would you like more details on the reasoning behind exponential graph transformations or have any specific questions?

Related Questions:

1. What is the general form of an exponential function and how do its parameters affect the graph?
2. How can you determine the vertical shift of a graph by analyzing its equation?
3. Why does the base of the exponential function influence whether the graph shows growth or decay?
4. How does horizontal translation (like $x - 1$) affect the graph of an exponential function?
5. Can a reflection of an exponential graph occur, and how would that affect its equation?

Tip:

When analyzing graphs of exponential functions, always check the y-intercept and the direction of the graph (whether it’s increasing or decreasing) to match it with its equation.