Let's analyze the graph of the function $y = f(x)$ provided in the image and answer each question step-by-step.

1. Find $f(2)$.

To find $f(2)$, locate $x = 2$ on the graph and observe the corresponding value of $y$.

• From the graph, when $x = 2$, $y = -1$.
  Therefore, $f(2) = -1$.

2. Find all the $x$-values such that $f(x) = 0$.

To find $x$-values where $f(x) = 0$, we look for points where the graph intersects the x-axis (where $y = 0$).

• The graph intersects the x-axis at the points $(-3, 0)$, $(0, 0)$, and $(3, 0)$.

  Therefore, $f(x) = 0$ when $x = -3, 0,$ and $3$.

3. Find the range of $f$.

The range of a function is the set of all possible values of $y$.

• The lowest point on the graph appears to be around $y = -3$ (occurring to the left of $x = -3$), and the highest point is at $y = 4$ when $x = 4$.

  Therefore, the range of $f$ is $[-3, 4]$.

4. Find all local extrema of $f$.

Local extrema are the local minimums and maximums on the graph.

• Local Maxima:

  • At $(-2, 1)$
  • At $(3, 0)$

• Local Minima:

  • At $(0, -1)$

5. Find all absolute extrema of $f$.

Absolute extrema refer to the highest and lowest values the function attains over its entire domain.

• Absolute Maximum: The absolute maximum value is $y = 4$ at $(4, 3)$.

• Absolute Minimum: The absolute minimum value is $y = -3$ occurring to the left beyond $x = -3$.

Would you like further details or have any other questions?

Further Questions:

1. What are the coordinates of other points on the graph that might be of interest?
2. How can you determine intervals where the function is increasing or decreasing?
3. What is the domain of the function $f$?
4. What are the critical points, and how are they identified?
5. How would this graph change if you shift it up or down by a certain value?

Tip: To identify extrema, look for points where the function changes direction, indicating local minimums or maximums.