Graphing a Parabola from Vertex Form Worksheet

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

Vertex Form:

$$
y = a(x - h)^2 + k
$$
- Vertex: $(h, k)$
- Axis of Symmetry (A.O.S.): $x = h$
- If $a > 0$: parabola opens upward, vertex is a minimum
- If $a < 0$: parabola opens downward, vertex is a maximum

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1. $ y = (x - 1)^2 + 2 $

- Compare to $ y = a(x - h)^2 + k $
- $ a = 1 $, $ h = 1 $, $ k = 2 $
- Vertex: $ (1, 2) $
- A.O.S.: $ x = 1 $
- Since $ a = 1 > 0 $ → opens up, so vertex is a minimum

✔ Answer:
- Vertex = $ (1, 2) $
- A.O.S. = $ x = 1 $
- Min

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2. $ y = 2(x - 2)^2 + 5 $

- $ a = 2 $, $ h = 2 $, $ k = 5 $
- Vertex: $ (2, 5) $
- A.O.S.: $ x = 2 $
- $ a = 2 > 0 $ → opens up, so vertex is a minimum

✔ Answer:
- Vertex = $ (2, 5) $
- A.O.S. = $ x = 2 $
- Min

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3. $ y = -3(x + 7)^2 - 8 $

Note: $ x + 7 = x - (-7) $, so $ h = -7 $

- $ a = -3 $, $ h = -7 $, $ k = -8 $
- Vertex: $ (-7, -8) $
- A.O.S.: $ x = -7 $
- $ a = -3 < 0 $ → opens down, so vertex is a maximum

✔ Answer:
- Vertex = $ (-7, -8) $
- A.O.S. = $ x = -7 $
- Max

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4. $ y = (x - 5)^2 - 3 $

- $ a = 1 $, $ h = 5 $, $ k = -3 $
- Vertex: $ (5, -3) $
- A.O.S.: $ x = 5 $
- $ a = 1 > 0 $ → opens up, so vertex is a minimum

✔ Answer:
- Vertex = $ (5, -3) $
- A.O.S. = $ x = 5 $
- Min

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Summary Table:

| Problem | Function | Vertex | A.O.S. | Max or Min |
|--------|---------|--------|--------|------------|
| 1 | $ y = (x - 1)^2 + 2 $ | $ (1, 2) $ | $ x = 1 $ | Min |
| 2 | $ y = 2(x - 2)^2 + 5 $ | $ (2, 5) $ | $ x = 2 $ | Min |
| 3 | $ y = -3(x + 7)^2 - 8 $ | $ (-7, -8) $ | $ x = -7 $ | Max |
| 4 | $ y = (x - 5)^2 - 3 $ | $ (5, -3) $ | $ x = 5 $ | Min |

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Graphing Tips (for sketching):

- Plot the vertex.
- Use the axis of symmetry as a guide.
- Pick a few $ x $-values on either side of the vertex and compute $ y $.
- For example, for $ y = (x - 1)^2 + 2 $:
- $ x = 0 $: $ y = (0 - 1)^2 + 2 = 1 + 2 = 3 $
- $ x = 2 $: $ y = (2 - 1)^2 + 2 = 1 + 2 = 3 $
- $ x = 1 $: $ y = 2 $
- So points: $ (0, 3), (1, 2), (2, 3) $ — symmetric about $ x = 1 $

Repeat this process for each function.

Let me know if you'd like me to generate the graphs!