Math II worksheet on graphing parabolas in factored form, including problems to identify key features and sketch graphs.

Problem: Analyze and sketch the given parabolas in factored form.

We are tasked with identifying the vertex, axis of symmetry, direction of opening, x-intercepts, and y-intercept for each parabola, and then sketching the graph. The equations are provided in factored form.

#### General Approach:
1. Factored Form: A quadratic function in factored form is written as:
\[
f(x) = a(x - r_1)(x - r_2)
\]
where \( r_1 \) and \( r_2 \) are the roots (x-intercepts).

2. Vertex: The x-coordinate of the vertex is the midpoint of the roots:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2}
\]
Substitute this value into the function to find the y-coordinate of the vertex.

3. Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex:
\[
x = x_{\text{vertex}}
\]

4. Direction of Opening:
- If \( a > 0 \), the parabola opens upwards.
- If \( a < 0 \), the parabola opens downwards.

5. x-Intercepts: These are the roots \( r_1 \) and \( r_2 \).

6. y-Intercept: Set \( x = 0 \) in the function to find the y-intercept.

7. Sketching: Use the vertex, axis of symmetry, x-intercepts, and y-intercept to sketch the parabola.

---

1. \( f(x) = -(x - 1)(x + 3) \)

#### Step 1: Identify the roots
The factored form is \( f(x) = -(x - 1)(x + 3) \). The roots are:
\[
r_1 = 1 \quad \text{and} \quad r_2 = -3
\]

#### Step 2: Find the vertex
The x-coordinate of the vertex is the midpoint of the roots:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2} = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1
\]
Substitute \( x = -1 \) into the function to find the y-coordinate:
\[
f(-1) = -((-1) - 1)((-1) + 3) = -(-2)(2) = -(-4) = 4
\]
Thus, the vertex is:
\[
(-1, 4)
\]

#### Step 3: Axis of Symmetry
The axis of symmetry is:
\[
x = -1
\]

#### Step 4: Direction of Opening
The coefficient \( a = -1 \) (from the leading term \(-1\)). Since \( a < 0 \), the parabola opens downwards.

#### Step 5: x-Intercepts
The x-intercepts are the roots:
\[
(1, 0) \quad \text{and} \quad (-3, 0)
\]

#### Step 6: y-Intercept
Set \( x = 0 \) in the function:
\[
f(0) = -(0 - 1)(0 + 3) = -(-1)(3) = 3
\]
Thus, the y-intercept is:
\[
(0, 3)
\]

#### Summary for \( f(x) = -(x - 1)(x + 3) \):
- Vertex: \((-1, 4)\)
- Axis of Symmetry: \( x = -1 \)
- Direction of Opening: Downwards
- x-Intercepts: \((1, 0)\) and \((-3, 0)\)
- y-Intercept: \((0, 3)\)

#### Sketch:
- Plot the vertex \((-1, 4)\).
- Draw the axis of symmetry \( x = -1 \).
- Mark the x-intercepts \((1, 0)\) and \((-3, 0)\).
- Mark the y-intercept \((0, 3)\).
- Sketch the parabola opening downwards.

---

2. \( f(x) = -\frac{1}{3}(x - 9)(x + 3) \)

#### Step 1: Identify the roots
The factored form is \( f(x) = -\frac{1}{3}(x - 9)(x + 3) \). The roots are:
\[
r_1 = 9 \quad \text{and} \quad r_2 = -3
\]

#### Step 2: Find the vertex
The x-coordinate of the vertex is the midpoint of the roots:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2} = \frac{9 + (-3)}{2} = \frac{6}{2} = 3
\]
Substitute \( x = 3 \) into the function to find the y-coordinate:
\[
f(3) = -\frac{1}{3}((3) - 9)((3) + 3) = -\frac{1}{3}(-6)(6) = -\frac{1}{3}(-36) = 12
\]
Thus, the vertex is:
\[
(3, 12)
\]

#### Step 3: Axis of Symmetry
The axis of symmetry is:
\[
x = 3
\]

#### Step 4: Direction of Opening
The coefficient \( a = -\frac{1}{3} \). Since \( a < 0 \), the parabola opens downwards.

#### Step 5: x-Intercepts
The x-intercepts are the roots:
\[
(9, 0) \quad \text{and} \quad (-3, 0)
\]

#### Step 6: y-Intercept
Set \( x = 0 \) in the function:
\[
f(0) = -\frac{1}{3}((0) - 9)((0) + 3) = -\frac{1}{3}(-9)(3) = -\frac{1}{3}(-27) = 9
\]
Thus, the y-intercept is:
\[
(0, 9)
\]

#### Summary for \( f(x) = -\frac{1}{3}(x - 9)(x + 3) \):
- Vertex: \((3, 12)\)
- Axis of Symmetry: \( x = 3 \)
- Direction of Opening: Downwards
- x-Intercepts: \((9, 0)\) and \((-3, 0)\)
- y-Intercept: \((0, 9)\)

#### Sketch:
- Plot the vertex \((3, 12)\).
- Draw the axis of symmetry \( x = 3 \).
- Mark the x-intercepts \((9, 0)\) and \((-3, 0)\).
- Mark the y-intercept \((0, 9)\).
- Sketch the parabola opening downwards.

---

3. \( f(x) = (x - 3)(x + 1) \)

#### Step 1: Identify the roots
The factored form is \( f(x) = (x - 3)(x + 1) \). The roots are:
\[
r_1 = 3 \quad \text{and} \quad r_2 = -1
\]

#### Step 2: Find the vertex
The x-coordinate of the vertex is the midpoint of the roots:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2} = \frac{3 + (-1)}{2} = \frac{2}{2} = 1
\]
Substitute \( x = 1 \) into the function to find the y-coordinate:
\[
f(1) = ((1) - 3)((1) + 1) = (-2)(2) = -4
\]
Thus, the vertex is:
\[
(1, -4)
\]

#### Step 3: Axis of Symmetry
The axis of symmetry is:
\[
x = 1
\]

#### Step 4: Direction of Opening
The coefficient \( a = 1 \) (from the leading term \(1\)). Since \( a > 0 \), the parabola opens upwards.

#### Step 5: x-Intercepts
The x-intercepts are the roots:
\[
(3, 0) \quad \text{and} \quad (-1, 0)
\]

#### Step 6: y-Intercept
Set \( x = 0 \) in the function:
\[
f(0) = ((0) - 3)((0) + 1) = (-3)(1) = -3
\]
Thus, the y-intercept is:
\[
(0, -3)
\]

#### Summary for \( f(x) = (x - 3)(x + 1) \):
- Vertex: \((1, -4)\)
- Axis of Symmetry: \( x = 1 \)
- Direction of Opening: Upwards
- x-Intercepts: \((3, 0)\) and \((-1, 0)\)
- y-Intercept: \((0, -3)\)

#### Sketch:
- Plot the vertex \((1, -4)\).
- Draw the axis of symmetry \( x = 1 \).
- Mark the x-intercepts \((3, 0)\) and \((-1, 0)\).
- Mark the y-intercept \((0, -3)\).
- Sketch the parabola opening upwards.

---

4. \( f(x) = -(4x - 8)(x + 2) \)

#### Step 1: Rewrite in standard factored form
First, factor out constants from each term:
\[
f(x) = -(4(x - 2))(x + 2) = -4(x - 2)(x + 2)
\]
Now the factored form is \( f(x) = -4(x - 2)(x + 2) \). The roots are:
\[
r_1 = 2 \quad \text{and} \quad r_2 = -2
\]

#### Step 2: Find the vertex
The x-coordinate of the vertex is the midpoint of the roots:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2} = \frac{2 + (-2)}{2} = \frac{0}{2} = 0
\]
Substitute \( x = 0 \) into the function to find the y-coordinate:
\[
f(0) = -4((0) - 2)((0) + 2) = -4(-2)(2) = -4(-4) = 16
\]
Thus, the vertex is:
\[
(0, 16)
\]

#### Step 3: Axis of Symmetry
The axis of symmetry is:
\[
x = 0
\]

#### Step 4: Direction of Opening
The coefficient \( a = -4 \). Since \( a < 0 \), the parabola opens downwards.

#### Step 5: x-Intercepts
The x-intercepts are the roots:
\[
(2, 0) \quad \text{and} \quad (-2, 0)
\]

#### Step 6: y-Intercept
Set \( x = 0 \) in the function:
\[
f(0) = -4((0) - 2)((0) + 2) = -4(-2)(2) = -4(-4) = 16
\]
Thus, the y-intercept is:
\[
(0, 16)
\]

#### Summary for \( f(x) = -(4x - 8)(x + 2) \):
- Vertex: \((0, 16)\)
- Axis of Symmetry: \( x = 0 \)
- Direction of Opening: Downwards
- x-Intercepts: \((2, 0)\) and \((-2, 0)\)
- y-Intercept: \((0, 16)\)

#### Sketch:
- Plot the vertex \((0, 16)\).
- Draw the axis of symmetry \( x = 0 \).
- Mark the x-intercepts \((2, 0)\) and \((-2, 0)\).
- Mark the y-intercept \((0, 16)\).
- Sketch the parabola opening downwards.

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
1) & \text{Vertex: } (-1, 4), \text{ Axis: } x = -1, \text{ Opens: Down}, \text{ x-Intercepts: } (1, 0), (-3, 0), \text{ y-Intercept: } (0, 3) \\
2) & \text{Vertex: } (3, 12), \text{ Axis: } x = 3, \text{ Opens: Down}, \text{ x-Intercepts: } (9, 0), (-3, 0), \text{ y-Intercept: } (0, 9) \\
3) & \text{Vertex: } (1, -4), \text{ Axis: } x = 1, \text{ Opens: Up}, \text{ x-Intercepts: } (3, 0), (-1, 0), \text{ y-Intercept: } (0, -3) \\
4) & \text{Vertex: } (0, 16), \text{ Axis: } x = 0, \text{ Opens: Down}, \text{ x-Intercepts: } (2, 0), (-2, 0), \text{ y-Intercept: } (0, 16) \\
\end{array}
}
\]