Let's analyze the graph and answer the questions:

a. Determine a piecewise equation for $f(x)$.

The graph consists of three distinct parts:

1. A parabola (a quadratic function) from $x = -3$ to $x = 2$.
2. A linear segment (a constant function) from $x = 2$ to $x = 4$.

Step 1: Parabola

The parabola appears to open upward and passes through the points $(-3, 3)$, $(0, 0)$, and $(2, -1)$. Let's assume it is of the form: $f(x) = ax^2 + bx + c$ Using the points $(-3, 3)$, $(0, 0)$, and $(2, -1)$, we can solve for $a$, $b$, and $c$.

• At $x = 0$, $f(0) = 0$ gives $c = 0$, so the equation becomes $f(x) = ax^2 + bx$.
• At $x = -3$, $f(-3) = 3$, we get $9a - 3b = 3$ → $3a - b = 1$ (Equation 1).
• At $x = 2$, $f(2) = -1$, we get $4a + 2b = -1$ (Equation 2).

Solving this system of equations: From Equation 1: $b = 3a - 1$. Substitute into Equation 2: $4a + 2(3a - 1) = -1$ $4a + 6a - 2 = -1$ $10a = 1 \quad \Rightarrow \quad a = \frac{1}{10}$ Substitute $a = \frac{1}{10}$ into $b = 3a - 1$: $b = 3\left(\frac{1}{10}\right) - 1 = \frac{3}{10} - 1 = \frac{-7}{10}$

Thus, the equation of the parabola is: $f(x) = \frac{1}{10}x^2 - \frac{7}{10}x$ This is valid for $-3 \leq x \leq 2$.

Step 2: Constant Line

The constant part is a horizontal line from $x = 2$ to $x = 4$, where $f(x) = -1$.

Thus, the piecewise function for $f(x)$ is: