Algebra II worksheet focusing on quadratic functions, covering standard and vertex forms, vertex identification, and graphing parabolas.

Let's go through the Algebra II: 5.2 Vertex Form Worksheet step by step and solve each part with clear explanations.

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🔹 Part 1: Tell whether the quadratic function is in standard form or vertex form

Standard form: $ y = ax^2 + bx + c $
Vertex form: $ y = a(x - h)^2 + k $

| Problem | Equation | Form |
|--------|----------|------|
| 1. $ y = x^2 - 2x - 35 $ | This is $ ax^2 + bx + c $ → standard ✔ |
| 2. $ y = 3(x - 1)^2 + 3 $ | This is $ a(x - h)^2 + k $ → vertex ✔ |
| 3. $ y = -\frac{2}{3}(x - 4)^2 + 7 $ | Same as above → vertex ✔ |
| 4. $ y = -2x^2 + 16x - 24 $ | This is $ ax^2 + bx + c $ → standard ✔ |

✔ All correct as filled.

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🔹 Part 2: Identify the vertex of the quadratic function in VERTEX form

For vertex form: $ y = a(x - h)^2 + k $, the vertex is (h, k)

| Problem | Equation | Vertex |
|--------|----------|--------|
| 5. $ y = 3(x - 7)^2 - 1 $ | $ h = 7, k = -1 $ → (7, -1) ✔ |
| 6. $ y = 3(x + 2)^2 - 5 $ | $ x + 2 = x - (-2) $ → $ h = -2, k = -5 $ → (-2, -5) ✔ |
| 7. $ y = (x - 3)^2 $ | $ h = 3, k = 0 $ → (3, 0) ✔ |
| 8. $ y = -4(x - 2)^2 + 4 $ | $ h = 2, k = 4 $ → (2, 4) ✔ |
| 9. $ y = 2(x + 1)^2 - 3 $ | $ h = -1, k = -3 $ → (-1, -3) ✔ |
| 10. $ y = (x + 4)^2 $ | $ h = -4, k = 0 $ → (-4, 0) ✔ |
| 11. $ y = \frac{1}{2}(x - 5)^2 + 1 $ | $ h = 5, k = 1 $ → (5, 1) ✔ |
| 12. $ y = -(x + 6)^2 + 10 $ | $ h = -6, k = 10 $ → (-6, 10) ✔ |

All vertices are correctly identified.

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🔹 Part 3: Identify the vertex of the quadratic function in STANDARD form

Use formula: $ x = -\frac{b}{2a} $, then plug into equation to find $ y $.

#### 13. $ y = 2x^2 - 16x + 31 $

- $ a = 2, b = -16 $
- $ x = -\frac{-16}{2(2)} = \frac{16}{4} = 4 $
- Plug $ x = 4 $:
$ y = 2(4)^2 - 16(4) + 31 = 2(16) - 64 + 31 = 32 - 64 + 31 = -1 $
- Vertex: $ (4, -1) $

✔ Given answer: $ y = 2(x - 4)^2 - 1 $ → matches vertex form → vertex is $ (4, -1) $

#### 14. $ y = -x^2 - 4x + 1 $

- $ a = -1, b = -4 $
- $ x = -\frac{-4}{2(-1)} = \frac{4}{-2} = -2 $
- Plug $ x = -2 $:
$ y = -(-2)^2 - 4(-2) + 1 = -4 + 8 + 1 = 5 $
- Vertex: $ (-2, 5) $

✔ Given answer: $ y = -(x + 2)^2 + 5 $ → $ (h,k) = (-2,5) $ → Correct

#### 15. $ y = 3x^2 - 6x + 4 $

- $ a = 3, b = -6 $
- $ x = -\frac{-6}{2(3)} = \frac{6}{6} = 1 $
- Plug $ x = 1 $:
$ y = 3(1)^2 - 6(1) + 4 = 3 - 6 + 4 = 1 $
- Vertex: $ (1, 1) $

✔ Given answer: $ y = 3(x - 1)^2 + 1 $ → matches → vertex $ (1,1) $

All correct!

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🔹 Part 4: Analyze and graph the given quadratic in vertex form

Given: $ y = -2(x + 2)^2 + 4 $

We analyze:

#### ✔ Vertex:
From $ y = a(x - h)^2 + k $, here:
- $ x + 2 = x - (-2) $ → $ h = -2 $
- $ k = 4 $
→ Vertex: (-2, 4) ✔

#### ✔ Axis of Symmetry:
$ x = h = -2 $ → x = -2 ✔

#### ✔ Opens:
Coefficient $ a = -2 $ → negative → opens down ✔

#### ✔ Maximum or Minimum:
Since it opens down → has a maximum value ✔

#### ✔ Max/Min Value:
The maximum value is the y-coordinate of the vertex: $ y = 4 $
So max value is 4, at point $ (-2, 4) $ ✔

#### ✔ y-intercept:
Set $ x = 0 $:
$ y = -2(0 + 2)^2 + 4 = -2(4) + 4 = -8 + 4 = -4 $
So y-intercept: (0, -4) ✘

But the worksheet says (0,2) — this is incorrect.

Wait! Let's double-check:

$$
y = -2(x + 2)^2 + 4
$$
At $ x = 0 $:
$$
y = -2(2)^2 + 4 = -2(4) + 4 = -8 + 4 = -4
$$
So y-intercept is (0, -4), not (0,2).

✘ So the student’s answer (0,2) is wrong.

✔ Correct y-intercept: $ (0, -4) $

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🔹 Graphing

The graph should show:

- Parabola opening downward
- Vertex at $ (-2, 4) $
- Axis of symmetry: vertical line $ x = -2 $
- y-intercept at $ (0, -4) $
- Additional points can be found by plugging in values:

| x | y = -2(x+2)² + 4 |
|---|------------------|
| -4 | -2(-2)² + 4 = -2(4)+4 = -4 |
| -3 | -2(-1)² + 4 = -2 + 4 = 2 |
| -2 | 4 → vertex |
| -1 | -2(1)² + 4 = -2 + 4 = 2 |
| 0 | -2(2)² + 4 = -8 + 4 = -4 |

So points: (-4,-4), (-3,2), (-2,4), (-1,2), (0,-4)

Plot these and draw a downward-opening parabola.

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✔ Final Answers Summary:

#### Part 1: Form Identification
1. Standard
2. Vertex
3. Vertex
4. Standard

#### Part 2: Vertex from Vertex Form
5. (7, -1)
6. (-2, -5)
7. (3, 0)
8. (2, 4)
9. (-1, -3)
10. (-4, 0)
11. (5, 1)
12. (-6, 10)

#### Part 3: Vertex from Standard Form
13. (4, -1)
14. (-2, 5)
15. (1, 1)

#### Part 4: Analysis of $ y = -2(x + 2)^2 + 4 $
- Vertex: (-2, 4) ✔
- Axis of symmetry: $ x = -2 $ ✔
- Opens: Down ✔
- Max/Min: Maximum ✔
- Max/Min Value: 4 ✔
- y-intercept: $ (0, -4) $ ✘ (not (0,2)) ← Correction needed

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📌 Final Note:

The y-intercept was incorrectly stated as (0,2) in the worksheet. It should be (0, -4).

✔ All other answers on the worksheet are correct.

Let me know if you'd like a hand-drawn version of the graph or help converting any standard form to vertex form!