Practice worksheet for graphing quadratic functions in intercept form, featuring three equations with corresponding graphs and key points labeled.

Let's go through each of the three quadratic functions given in intercept form and verify the solutions provided, then explain how to solve them step by step.

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🔷 General Form:

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

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## ✔ Problem 1:
$$
y = \frac{1}{2}(x + 4)(x - 2)
$$

Step 1: X-intercepts

Set $y = 0$:
$$
\frac{1}{2}(x + 4)(x - 2) = 0 \Rightarrow x = -4, x = 2
$$
✔ So, x-intercepts: $(-4, 0)$ and $(2, 0)$ — correct

Step 2: Axis of Symmetry

Midpoint of $-4$ and $2$:
$$
x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1
$$
✔ Axis of symmetry: $x = -1$ — correct

Step 3: Vertex

Plug $x = -1$ into the equation:
$$
y = \frac{1}{2}(-1 + 4)(-1 - 2) = \frac{1}{2}(3)(-3) = \frac{1}{2}(-9) = -4.5
$$
So vertex is $(-1, -4.5)$ — ✔ correct

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: $(0, -4)$ — ✔ correct

✔️ All answers for #1 are correct.

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## ✔ Problem 2:
$$
y = -\frac{1}{2}x(x - 8)
$$

Step 1: X-intercepts

Set $y = 0$:
$$
-\frac{1}{2}x(x - 8) = 0 \Rightarrow x = 0, x = 8
$$
✔ So x-intercepts: $(0, 0)$ and $(8, 0)$ — correct

Step 2: Axis of Symmetry

Midpoint of $0$ and $8$:
$$
x = \frac{0 + 8}{2} = 4
$$
✔ Axis of symmetry: $x = 4$ — correct

Step 3: Vertex

Plug $x = 4$:
$$
y = -\frac{1}{2}(4)(4 - 8) = -\frac{1}{2}(4)(-4) = -\frac{1}{2}(-16) = 8
$$
So vertex: $(4, 8)$ — ✔ correct

Step 4: Y-intercept

Set $x = 0$:
$$
y = -\frac{1}{2}(0)(0 - 8) = 0
$$
So y-intercept: $(0, 0)$ — ✔ correct

✔️ All answers for #2 are correct.

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## ✔ Problem 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: $(-2, 0)$ and $(2, 0)$ — correct

Step 2: Axis of Symmetry

Midpoint of $-2$ and $2$:
$$
x = \frac{-2 + 2}{2} = 0
$$
✔ Axis of symmetry: $x = 0$ — correct

Step 3: Vertex

Plug $x = 0$:
$$
y = (0 + 2)(0 - 2) = (2)(-2) = -4
$$
So vertex: $(0, -4)$ — ✔ correct

Step 4: Y-intercept

Set $x = 0$:
$$
y = (0 + 2)(0 - 2) = -4
$$
So y-intercept: $(0, -4)$ — ✔ correct

✔️ All answers for #3 are correct.

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📌 Final Summary:

All the filled-in answers on your worksheet are correct!

Here’s a quick recap:

| Function | X-ints | Axis of Symmetry | Vertex | Y-int |
|--------|--------|------------------|--------|-------|
| $y = \frac{1}{2}(x+4)(x-2)$ | $(-4,0), (2,0)$ | $x = -1$ | $(-1, -4.5)$ | $(0, -4)$ |
| $y = -\frac{1}{2}x(x-8)$ | $(0,0), (8,0)$ | $x = 4$ | $(4, 8)$ | $(0, 0)$ |
| $y = (x+2)(x-2)$ | $(-2,0), (2,0)$ | $x = 0$ | $(0, -4)$ | $(0, -4)$ |

You’ve done an excellent job! Each graph should reflect these key points, with parabolas opening upward if $a > 0$, downward if $a < 0$. For example:
- #1 and #3 open upward (positive leading coefficient),
- #2 opens downward (negative coefficient).

Let me know if you'd like help sketching or plotting any of these graphs!