Practice worksheet for graphing quadratic functions in intercept form, including identifying vertex, axis of symmetry, and direction of opening.

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, and y-intercept. Draw a smooth U-shaped curve.

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

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

Axis of symmetry:
$$
x = \frac{1 + 6}{2} = 3.5
$$

Vertex: Plug $x = 3.5$
$$
y = -(3.5 - 1)(3.5 - 6) = -(2.5)(-2.5) = 6.25
$$
✔ Vertex: $(3.5, 6.25)$

Direction: $a = -1 < 0$, so opens downward

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

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#### 3) $ y = 2(x - 2)(x - 5) $

x-intercepts: $x = 2$, $x = 5$

Axis of symmetry: $x = \frac{2+5}{2} = 3.5$

Vertex: $x = 3.5$
$$
y = 2(3.5 - 2)(3.5 - 5) = 2(1.5)(-1.5) = 2(-2.25) = -4.5
$$
✔ Vertex: $(3.5, -4.5)$

Opens: $a = 2 > 0$ → upward

y-intercept: $x = 0$
$$
y = 2(-2)(-5) = 2(10) = 20
$$
✔ y-intercept: $(0, 20)$

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

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

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

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

Opens: $a = -2 < 0$ → downward

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

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#### 5) $ y = -2(x - 1)(x - 3) $

x-intercepts: $x = 1$, $x = 3$

Axis of symmetry: $x = \frac{1 + 3}{2} = 2$

Vertex: $x = 2$
$$
y = -2(2 - 1)(2 - 3) = -2(1)(-1) = 2
$$
✔ Vertex: $(2, 2)$

Opens: $a = -2 < 0$ → downward

y-intercept: $x = 0$
$$
y = -2(-1)(-3) = -2(3) = -6
$$
✔ y-intercept: $(0, -6)$

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✔ Summary Table (for your worksheet)

| Equation | x-ints | Axis of Symmetry | Vertex | Opens | 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) | $x = 3.5$ | (3.5, 6.25) | Down | (0, -6) |
| $y = 2(x - 2)(x - 5)$ | (2,0), (5,0) | $x = 3.5$ | (3.5, -4.5) | Up | (0, 20) |
| $y = -2(x + 3)(x - 2)$ | (-3,0), (2,0) | $x = -0.5$ | (-0.5, 12.5) | Down | (0, 12) |
| $y = -2(x - 1)(x - 3)$ | (1,0), (3,0) | $x = 2$ | (2, 2) | Down | (0, -6) |

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📝 How to Fill in Your Worksheet:

For each problem:
1. Write x-intercepts as ordered pairs.
2. Write axis of symmetry as $x = \text{value}$.
3. Compute and write vertex as $(x, y)$.
4. Determine if it opens up or down based on sign of $a$.
5. Compute y-intercept by plugging $x = 0$.
6. Sketch the graph using these points.

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If you can type out one of the problems from the image (or describe it), I can help you solve that specific one with full detail!