Let's work through this problem step by step. The x-intercepts of the graph of $y = f(x)$ are given as $-5$ and $3$. This means that $f(-5) = 0$ and $f(3) = 0$, which are the points where the graph crosses the x-axis.

We will now answer the given questions based on transformations of $f(x)$.

(a) What are the x-intercepts of the graph of $y = f(x + 6)$?

• The transformation $f(x + 6)$ represents a horizontal shift to the left by 6 units.
• The original x-intercepts are at $-5$ and $3$. After shifting left by 6, the new intercepts are: $-5 - 6 = -11 \quad \text{and} \quad 3 - 6 = -3.$ Answer: The x-intercepts are $-11$ and $-3$.

(b) What are the x-intercepts of the graph of $y = f(x - 8)$?

• The transformation $f(x - 8)$ represents a horizontal shift to the right by 8 units.
• The original x-intercepts are at $-5$ and $3$. After shifting right by 8, the new intercepts are: $-5 + 8 = 3 \quad \text{and} \quad 3 + 8 = 11.$ Answer: The x-intercepts are $3$ and $11$.

(c) What are the x-intercepts of the graph of $y = 2f(x)$?

• The transformation $2f(x)$ represents a vertical stretch by a factor of 2.
• A vertical stretch does not change the x-intercepts, as the graph is only stretched vertically, leaving the points where $f(x) = 0$ unchanged. $\text{The x-intercepts remain the same at } -5 \text{ and } 3.$ Answer: The x-intercepts are $-5$ and $3$.

(d) What are the x-intercepts of the graph of $y = f(-x)$?

• The transformation $f(-x)$ represents a reflection over the y-axis.
• To find the new x-intercepts, we reflect the original x-intercepts $-5$ and $3$ over the y-axis. This means we take the opposite signs: $-(-5) = 5 \quad \text{and} \quad -(3) = -3.$ Answer: The x-intercepts are $5$ and $-3$.

Would you like further clarification or additional details? Here are some related questions:

1. How does the transformation $y = f(kx)$ affect the graph's x-intercepts?
2. Can you explain how vertical and horizontal transformations differ?
3. What happens to the x-intercepts if we apply a transformation like $y = f(x) + k$?
4. How do reflections over the x-axis affect the x-intercepts of a function?
5. If a graph is shifted both vertically and horizontally, how do we determine the new x-intercepts?

Tip: When working with function transformations, remember that horizontal shifts affect the x-values, while vertical shifts affect the y-values.