https://file.aiquickdraw.com/mathbot/file/1727018325359e3dv1qp9.jpg

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:
\[
f(x) =
\begin{cases} 
\frac{1}{10}x^2 - \frac{7}{10}x, & \text{for } -3 \leq x \leq 2 \\
-1, & \text{for } 2 < x \leq 4 
\end{cases}
\]

---

### b. Determine the interval(s) where \( f(x) \) is increasing.

The graph shows that \( f(x) \) is increasing on the interval:
\[
(0, 2]
\]

---

### c. Determine the interval(s) where \( f(x) \) is decreasing.

The graph shows that \( f(x) \) is decreasing on the interval:
\[
[-3, 0]
\]

---

### d. Determine the interval(s) where \( f(x) \) is constant.

The graph shows that \( f(x) \) is constant on the interval:
\[
(2, 4]
\]

---

Would you like further details or clarifications on any of these points?

### 5 Related Questions:
1. How do we find the derivative of the quadratic equation \( f(x) = \frac{1}{10}x^2 - \frac{7}{10}x \)?
2. What are the critical points of the function \( f(x) \) on the interval \( [-3, 4] \)?
3. How do we determine concavity from a graph or an equation?
4. How do piecewise functions behave when differentiating at the boundary points?
5. What is the significance of intervals where a function is increasing or decreasing?

### Tip:
When identifying increasing or decreasing intervals, it's helpful to look for turning points where the slope changes from positive to negative or vice versa (like local maxima or minima).

The function f(x) is graphed below. Use the graph to answer the following: a) Determine a piecewise equation for f(x), b) Determine the interval(s) where f(x) is increasing, c) Determine the interval(s) where f(x) is decreasing, d) Determine the interval(s) where f(x) is constant.

Explore the graph of a piecewise function to determine the quadratic equation and identify intervals where the function is increasing, decreasing, or constant. This problem demonstrates the use of algebra and graph interpretation, suitable for high school students.

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