The question involves analyzing the graph of $y = f(x)$ and determining various values for $f(x)$ based on the graph. I'll provide step-by-step help with each part shown in the image:

Part 1 of 8

(a) Determine $f(-4)$:

• The value $f(-4)$ represents the y-coordinate of the point on the graph where $x = -4$. From the graph, it appears that when $x = -4$, the value of $y = 0$. Therefore: $f(-4) = 0$

Part 2 of 8

(b) Determine $f(1)$:

• Similarly, $f(1)$ is the y-coordinate when $x = 1$. The graph shows that when $x = 1$, $y = -1$. Hence: $f(1) = -1$

Part 3 of 8

(c) Find all $x$ for which $f(x) = -1$:

• The question asks for the values of $x$ where the function equals $-1$. By looking at the graph, we can see that the function intersects $y = -1$ at $x = -3, 1,$ and $3$. Thus: $\text{Values of } x \text{ for which } f(x) = -1: \quad x = -3, 1, 3$

Part 4 of 8

(d) Find all $x$ for which $f(x) = -4$:

• From the graph, we look for the points where the function equals $-4$. There seems to be no point on the graph where $y = -4$, so: $\text{There is no } x \text{ such that } f(x) = -4.$

Let me know if you need any additional details or clarification on these steps.

Here are 5 related questions to expand on this topic:

1. What is the value of $f(0)$ from the graph?
2. For what value(s) of $x$ does $f(x) = 0$?
3. What is the maximum value of $f(x)$ on the graph?
4. Find all values of $x$ where the graph has local minima.
5. How can you tell if a function is increasing or decreasing from its graph?

Tip: When analyzing graphs, always check the y-coordinates corresponding to the given x-values carefully, especially at intersection points.