To graph the function $y = 3^x - 3$, let's examine its behavior and identify which of the options matches its characteristics.

Step 1: Analyze the function $y = 3^x - 3$

1. Base Function: $y = 3^x$ is an exponential function that grows rapidly for positive $x$ and approaches zero as $x$ becomes negative.
2. Vertical Shift: The $-3$ subtracts 3 from every point on $y = 3^x$, shifting the entire graph downward by 3 units.

Step 2: Key Points Calculation

To identify the correct graph, we can calculate specific points:

• For $x = -2$: $y = 3^{-2} - 3 = \frac{1}{9} - 3 \approx -2.89$
• For $x = 0$: $y = 3^0 - 3 = 1 - 3 = -2$
• For $x = 2$: $y = 3^2 - 3 = 9 - 3 = 6$

Step 3: Match with Options

From the options given, we look for a graph with these approximate points:

• $(-2, \approx -3)$
• $(0, -2)$
• $(2, 6)$

Answer

Based on these points, the correct answer is: Option C, which matches the values and the pattern of points calculated for the function $y = 3^x - 3$.

Would you like a detailed plot of this function, or do you have any other questions?

Related Questions

1. How does the base of the exponential function affect the graph shape?
2. What changes would occur if the vertical shift were positive?
3. How does changing the base of the exponent affect the rate of growth?
4. What would happen to the graph if a horizontal shift were added?
5. How would this graph differ if the function were $y = 3^{-x} - 3$?

Tip

When dealing with exponential functions, always consider transformations (like shifts and stretches) to determine how the graph is modified.