Let’s analyze the graph and answer each of the following questions.

(a) Intercepts

• x-intercepts: Points where the function crosses the x-axis. From the graph, the function crosses the x-axis at $x = -2$ and $x = 2$. So the x-intercepts are $(-2, 0)$ and $(2, 0)$.
• y-intercept: The point where the function crosses the y-axis. In the graph, the function crosses the y-axis at $y = 2$, so the y-intercept is $(0, 2)$.

(b) Domain and Range

• Domain: The set of all x-values for which the function is defined. The graph spans from $x = -4$ to $x = 4$. Therefore, the domain is: $\text{Domain: } [-4, 4]$
• Range: The set of all y-values the function takes. From the graph, the lowest point is at $y = 0$ and the highest point is $y = 4$. Therefore, the range is: $\text{Range: } [0, 4]$

(c) Intervals of Increase, Decrease, and Constant

• Increasing: The function is increasing where the slope of the line is positive. From the graph:

  • Increasing from $x = -2$ to $x = 0$.
  • Increasing from $x = 2$ to $x = 4$.

• Decreasing: The function is decreasing where the slope is negative. From the graph:

  • Decreasing from $x = -4$ to $x = -2$.
  • Decreasing from $x = 0$ to $x = 2$.

• Constant: There are no intervals where the function is constant since the graph consists entirely of sloped lines.

(d) Symmetry: Even, Odd, or Neither

• Even function: A function is even if $f(x) = f(-x)$ for all x in the domain, meaning the graph is symmetric about the y-axis.
• Odd function: A function is odd if $f(-x) = -f(x)$, meaning the graph is symmetric about the origin.

From the graph, it is neither symmetric about the y-axis nor the origin. Hence, the function is neither even nor odd.

Would you like further clarification on any part?

Here are five related questions to expand your understanding:

1. How would the graph change if the function were even or odd?
2. What is the significance of the x- and y-intercepts in real-world applications?
3. Can the domain of a function ever be larger than what is shown on the graph?
4. How do you determine whether a piecewise function is increasing or decreasing?
5. How do you find the symmetry of more complex functions analytically?

Tip: Always check both symmetry and behavior at intercepts when analyzing the nature of a function graph.