Worksheet for graphing quadratic functions in intercept form with six equations, intercepts, symmetry, vertex, and y-intercept details.

Let's solve each problem step by step using the intercept form of a quadratic function:

The intercept form is:
$$
y = a(x - p)(x - q)
$$
where:
- $p$ and $q$ are the x-intercepts (roots),
- The axis of symmetry is the vertical line halfway between $p$ and $q$:
$$
x = \frac{p + q}{2}
$$
- The vertex lies on the axis of symmetry. Plug this $x$ into the equation to find the $y$-coordinate.
- The y-intercept is found by setting $x = 0$.

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1) $ y = \frac{1}{2}(x + 4)(x - 2) $

#### Step 1: X-intercepts
Set $y = 0$:
$$
0 = \frac{1}{2}(x + 4)(x - 2)
\Rightarrow x = -4, \quad x = 2
$$
So, x-intercepts: $(-4, 0)$, $(2, 0)$

#### Step 2: Axis of Symmetry
$$
x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1
$$
So, Axis of Symmetry: $x = -1$

#### Step 3: Vertex
Plug $x = -1$ into the equation:
$$
y = \frac{1}{2}(-1 + 4)(-1 - 2) = \frac{1}{2}(3)(-3) = \frac{1}{2}(-9) = -\frac{9}{2} = -4.5
$$
So, Vertex: $(-1, -4.5)$

#### Step 4: Y-intercept
Set $x = 0$:
$$
y = \frac{1}{2}(0 + 4)(0 - 2) = \frac{1}{2}(4)(-2) = \frac{1}{2}(-8) = -4
$$
So, Y-intercept: $(0, -4)$

#### Graphing Tip:
Plot the x-intercepts, vertex, and y-intercept. Use symmetry to sketch the parabola (opens up since $a = \frac{1}{2} > 0$).

---

2) $ y = -\frac{1}{2}(x - 8)(x - 8) $

Wait — this is:
$$
y = -\frac{1}{2}(x - 8)^2
$$
This is a perfect square, so it has a double root at $x = 8$.

#### Step 1: X-intercepts
Set $y = 0$:
$$
0 = -\frac{1}{2}(x - 8)^2 \Rightarrow x = 8
$$
So, x-intercepts: $(8, 0)$, $(8, 0)$ → just one point.

#### Step 2: Axis of Symmetry
Since both roots are at $x = 8$, axis is:
$$
x = 8
$$

#### Step 3: Vertex
At $x = 8$:
$$
y = -\frac{1}{2}(8 - 8)^2 = 0
$$
So, Vertex: $(8, 0)$

#### Step 4: Y-intercept
Set $x = 0$:
$$
y = -\frac{1}{2}(0 - 8)^2 = -\frac{1}{2}(64) = -32
$$
So, Y-intercept: $(0, -32)$

Note: This is a downward-opening parabola with vertex at $(8, 0)$.

---

3) $ y = (x + 2)(x - 2) $

This is a difference of squares: $y = x^2 - 4$

#### Step 1: X-intercepts
Set $y = 0$:
$$
(x + 2)(x - 2) = 0 \Rightarrow x = -2, x = 2
$$
So, x-intercepts: $(-2, 0)$, $(2, 0)$

#### Step 2: Axis of Symmetry
$$
x = \frac{-2 + 2}{2} = 0
$$
So, Axis of Symmetry: $x = 0$

#### Step 3: Vertex
Plug $x = 0$:
$$
y = (0 + 2)(0 - 2) = (2)(-2) = -4
$$
So, Vertex: $(0, -4)$

#### Step 4: Y-intercept
Set $x = 0$: same as above → $(0, -4)$

Note: This is a U-shaped parabola opening upward.

---

4) $ y = -\frac{1}{3}(x + 1)(x - 5) $

#### Step 1: X-intercepts
Set $y = 0$:
$$
x = -1, \quad x = 5
$$
So, x-intercepts: $(-1, 0)$, $(5, 0)$

#### Step 2: Axis of Symmetry
$$
x = \frac{-1 + 5}{2} = \frac{4}{2} = 2
$$
So, Axis of Symmetry: $x = 2$

#### Step 3: Vertex
Plug $x = 2$:
$$
y = -\frac{1}{3}(2 + 1)(2 - 5) = -\frac{1}{3}(3)(-3) = -\frac{1}{3}(-9) = 3
$$
So, Vertex: $(2, 3)$

#### Step 4: Y-intercept
Set $x = 0$:
$$
y = -\frac{1}{3}(0 + 1)(0 - 5) = -\frac{1}{3}(1)(-5) = \frac{5}{3} \approx 1.67
$$
So, Y-intercept: $(0, \frac{5}{3})$

Note: Opens downward ($a = -\frac{1}{3} < 0$)

---

5) $ y = 4(x + 2)(x + 1) $

#### Step 1: X-intercepts
Set $y = 0$:
$$
x = -2, \quad x = -1
$$
So, x-intercepts: $(-2, 0)$, $(-1, 0)$

#### Step 2: Axis of Symmetry
$$
x = \frac{-2 + (-1)}{2} = \frac{-3}{2} = -1.5
$$
So, Axis of Symmetry: $x = -1.5$

#### Step 3: Vertex
Plug $x = -1.5$:
$$
y = 4(-1.5 + 2)(-1.5 + 1) = 4(0.5)(-0.5) = 4(-0.25) = -1
$$
So, Vertex: $(-1.5, -1)$

#### Step 4: Y-intercept
Set $x = 0$:
$$
y = 4(0 + 2)(0 + 1) = 4(2)(1) = 8
$$
So, Y-intercept: $(0, 8)$

Opens upward ($a = 4 > 0$)

---

6) $ y = -(x - 3)(x - 3) = -(x - 3)^2 $

This is a perfect square.

#### Step 1: X-intercepts
Set $y = 0$:
$$
-(x - 3)^2 = 0 \Rightarrow x = 3
$$
So, x-intercepts: $(3, 0)$, $(3, 0)$

#### Step 2: Axis of Symmetry
$$
x = 3
$$

#### Step 3: Vertex
At $x = 3$: $y = -(3 - 3)^2 = 0$
So, Vertex: $(3, 0)$

#### Step 4: Y-intercept
Set $x = 0$:
$$
y = -(0 - 3)^2 = -9
$$
So, Y-intercept: $(0, -9)$

Downward-opening parabola.

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Final Answers Summary:

| Problem | x-intercepts | Axis of Symmetry | Vertex | y-intercept |
|--------|--------------|------------------|--------|-------------|
| 1) | (-4, 0), (2, 0) | $x = -1$ | (-1, -4.5) | (0, -4) |
| 2) | (8, 0), (8, 0) | $x = 8$ | (8, 0) | (0, -32) |
| 3) | (-2, 0), (2, 0) | $x = 0$ | (0, -4) | (0, -4) |
| 4) | (-1, 0), (5, 0) | $x = 2$ | (2, 3) | (0, 5/3) |
| 5) | (-2, 0), (-1, 0) | $x = -1.5$ | (-1.5, -1) | (0, 8) |
| 6) | (3, 0), (3, 0) | $x = 3$ | (3, 0) | (0, -9) |

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Graphing Instructions:

For each problem:
- Plot the x-intercepts.
- Draw the axis of symmetry as a dashed vertical line.
- Plot the vertex.
- Plot the y-intercept.
- Use symmetry to plot one more point (e.g., mirror of y-intercept across axis).
- Sketch the parabola based on direction (up or down).

Let me know if you'd like me to draw the graphs or explain how to sketch them!