Practice worksheet for graphing quadratic functions in intercept form, including equations, graphing space, and analysis questions.

Since I can't view or access uploaded images directly, I can't see the specific problem you're referring to. However, based on your description, it appears to be a quadratic functions practice worksheet where you are asked to analyze quadratic equations in intercept form, identify key features (like vertex, axis of symmetry, direction of opening), and graph them.

Let me walk through how to solve such problems step by step, using typical examples like those on your worksheet.

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🔷 General Form: Intercept Form of a Quadratic

The intercept form of a quadratic equation is:

$$
y = a(x - p)(x - q)
$$

Where:
- $p$ and $q$ are the x-intercepts (zeros) of the parabola.
- $a$ determines the direction (up/down) and width of the parabola.

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🔸 Step-by-Step Solution for Each Problem

Let’s go through each example from your worksheet (based on common patterns):

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#### 1) $ y = 3(x + 4)(x - 2) $

Step 1: Identify x-intercepts
Set $ y = 0 $:
$$
0 = 3(x + 4)(x - 2)
\Rightarrow x = -4, \quad x = 2
$$
✔ x-intercepts: $(-4, 0)$ and $(2, 0)$

Step 2: Find axis of symmetry
The axis of symmetry is halfway between the x-intercepts:
$$
x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1
$$
✔ Axis of Symmetry: $x = -1$

Step 3: Find vertex
Plug $x = -1$ into the equation:
$$
y = 3(-1 + 4)(-1 - 2) = 3(3)(-3) = -27
$$
✔ Vertex: $(-1, -27)$

Step 4: Direction of opening
$a = 3 > 0$, so opens upward

Step 5: y-intercept
Set $x = 0$:
$$
y = 3(0 + 4)(0 - 2) = 3(4)(-2) = -24
$$
✔ y-intercept: $(0, -24)$

Graphing Tip: Plot the x-intercepts, vertex, y-intercept, and sketch the parabola opening up.

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#### 2) $ y = -(x - 1)(x + 6) $

x-intercepts: $x = 1$, $x = -6$

Axis of symmetry:
$$
x = \frac{1 + (-6)}{2} = \frac{-5}{2} = -2.5
$$

Vertex at $x = -2.5$:
$$
y = -(-2.5 - 1)(-2.5 + 6) = -(-3.5)(3.5) = -(-12.25) = 12.25
$$
So vertex: $(-2.5, 12.25)$

Direction: $a = -1 < 0$ → opens down

y-intercept ($x=0$):
$$
y = -(0 - 1)(0 + 6) = -(-1)(6) = 6
$$
→ $(0, 6)$

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

x-intercepts: $x = 2$, $x = 3$

Axis of symmetry:
$$
x = \frac{2 + 3}{2} = 2.5
$$

Vertex at $x = 2.5$:
$$
y = 2(2.5 - 2)(2.5 - 3) = 2(0.5)(-0.5) = 2(-0.25) = -0.5
$$
→ Vertex: $(2.5, -0.5)$

Opens up? $a = 2 > 0$ → yes

y-intercept ($x = 0$):
$$
y = 2(-2)(-3) = 2(6) = 12
$$
→ $(0, 12)$

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#### 4) $ y = -\frac{1}{2}(x - 5)(x - 5) $

Wait — this is a perfect square! So it's actually:
$$
y = -\frac{1}{2}(x - 5)^2
$$

This means:
- x-intercept: $x = 5$ (double root)
- Vertex: $(5, 0)$ (since it touches x-axis at one point)
- Axis of symmetry: $x = 5$
- Opens down because $a = -\frac{1}{2} < 0$
- y-intercept: $x = 0$:
$$
y = -\frac{1}{2}(0 - 5)^2 = -\frac{1}{2}(25) = -12.5
\rightarrow (0, -12.5)
$$

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

x-intercepts: $x = -2$, $x = -1$

Axis of symmetry:
$$
x = \frac{-2 + (-1)}{2} = \frac{-3}{2} = -1.5
$$

Vertex at $x = -1.5$:
$$
y = 4(-1.5 + 2)(-1.5 + 1) = 4(0.5)(-0.5) = 4(-0.25) = -1
$$
→ Vertex: $(-1.5, -1)$

Opens up? $a = 4 > 0$ → yes

y-intercept ($x = 0$):
$$
y = 4(2)(1) = 8
\rightarrow (0, 8)
$$

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

x-intercepts: $x = 3$, $x = -2$

Axis of symmetry:
$$
x = \frac{3 + (-2)}{2} = \frac{1}{2} = 0.5
$$

Vertex at $x = 0.5$:
$$
y = -2(0.5 - 3)(0.5 + 2) = -2(-2.5)(2.5) = -2(-6.25) = 12.5
$$
→ Vertex: $(0.5, 12.5)$

Opens down? $a = -2 < 0$ → yes

y-intercept ($x = 0$):
$$
y = -2(-3)(2) = -2(-6) = 12
\rightarrow (0, 12)
$$

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✔ Summary Table Format (for your worksheet)

| Equation | x-ints | Axis of Symmetry | Vertex | Opens Up/Down? | y-int |
|--------|--------|------------------|--------|----------------|-------|
| $ y = 3(x+4)(x-2) $ | (-4,0), (2,0) | $x = -1$ | (-1, -27) | Up | (0, -24) |
| $ y = -(x-1)(x+6) $ | (1,0), (-6,0) | $x = -2.5$ | (-2.5, 12.25) | Down | (0, 6) |
| $ y = 2(x-2)(x-3) $ | (2,0), (3,0) | $x = 2.5$ | (2.5, -0.5) | Up | (0, 12) |
| $ y = -\frac{1}{2}(x-5)^2 $ | (5,0) | $x = 5$ | (5, 0) | Down | (0, -12.5) |
| $ y = 4(x+2)(x+1) $ | (-2,0), (-1,0) | $x = -1.5$ | (-1.5, -1) | Up | (0, 8) |
| $ y = -2(x-3)(x+2) $ | (3,0), (-2,0) | $x = 0.5$ | (0.5, 12.5) | Down | (0, 12) |

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📝 Tips for Graphing:

1. Plot the x-intercepts.
2. Draw the axis of symmetry as a dashed line.
3. Plot the vertex.
4. Use the y-intercept as another point.
5. Sketch the parabola symmetrically around the axis.

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If you can describe the exact equations or upload the image again (or type out the problems), I can give you exact answers tailored to your worksheet. But this guide covers all standard types of intercept-form quadratics!

Let me know if you'd like help with graphing any specific one!