Vertex Form of Parabolas - Kuta Software - Free Printable

Let's solve each problem by converting the given quadratic equations into vertex form. The vertex form of a parabola is:

$$
y = a(x - h)^2 + k
$$

where $(h, k)$ is the vertex of the parabola.

We’ll go through each problem one by one and complete the square (or simplify) to get it into vertex form.

---

1) $ y = x^2 + 16x + 71 $

Complete the square:

- Take half of 16: $ \frac{16}{2} = 8 $
- Square it: $ 8^2 = 64 $
- Add and subtract 64:

$$
y = x^2 + 16x + 64 - 64 + 71 = (x + 8)^2 + 7
$$

✔ Answer: $ y = (x + 8)^2 + 7 $

---

2) $ y = x^2 - 2x - 5 $

- Half of $-2$: $-1$, square: $1$
- Add and subtract 1:

$$
y = x^2 - 2x + 1 - 1 - 5 = (x - 1)^2 - 6
$$

✔ Answer: $ y = (x - 1)^2 - 6 $

---

3) $ y = -x^2 - 14x - 59 $

Factor out $-1$ from the first two terms:

$$
y = -(x^2 + 14x) - 59
$$

- Half of 14: 7, square: 49
- Add and subtract 49 inside parentheses:

$$
y = -(x^2 + 14x + 49 - 49) - 59 = -( (x + 7)^2 - 49 ) - 59
$$

Distribute the negative:

$$
y = -(x + 7)^2 + 49 - 59 = -(x + 7)^2 - 10
$$

✔ Answer: $ y = -(x + 7)^2 - 10 $

---

4) $ y = 2x^2 + 36x + 170 $

Factor out 2 from first two terms:

$$
y = 2(x^2 + 18x) + 170
$$

- Half of 18: 9, square: 81
- Add and subtract 81 inside parentheses:

$$
y = 2(x^2 + 18x + 81 - 81) + 170 = 2( (x + 9)^2 - 81 ) + 170
$$

Distribute:

$$
y = 2(x + 9)^2 - 162 + 170 = 2(x + 9)^2 + 8
$$

✔ Answer: $ y = 2(x + 9)^2 + 8 $

---

5) $ y = x^2 - 12x + 46 $

- Half of $-12$: $-6$, square: 36
- Add and subtract 36:

$$
y = x^2 - 12x + 36 - 36 + 46 = (x - 6)^2 + 10
$$

✔ Answer: $ y = (x - 6)^2 + 10 $

---

6) $ y = x^2 + 4x $

- Half of 4: 2, square: 4
- Add and subtract 4:

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

✔ Answer: $ y = (x + 2)^2 - 4 $

---

7) $ y = x^2 - 6x + 5 $

- Half of $-6$: $-3$, square: 9
- Add and subtract 9:

$$
y = x^2 - 6x + 9 - 9 + 5 = (x - 3)^2 - 4
$$

✔ Answer: $ y = (x - 3)^2 - 4 $

---

8) $ y = (x + 5)(x + 4) $

First expand:

$$
y = x^2 + 9x + 20
$$

Now complete the square:

- Half of 9: $4.5$, square: $20.25$
- Add and subtract:

$$
y = x^2 + 9x + 20.25 - 20.25 + 20 = (x + 4.5)^2 - 0.25
$$

Or write as fractions:

- $4.5 = \frac{9}{2}$, $20.25 = \left(\frac{9}{2}\right)^2 = \frac{81}{4}$

So:

$$
y = \left(x + \frac{9}{2}\right)^2 - \frac{81}{4} + \frac{80}{4} = \left(x + \frac{9}{2}\right)^2 - \frac{1}{4}
$$

✔ Answer: $ y = \left(x + \frac{9}{2}\right)^2 - \frac{1}{4} $

---

9) $ \frac{1}{2}(y + 4) = (x - 7)^2 $

Solve for $y$:

Multiply both sides by 2:

$$
y + 4 = 2(x - 7)^2
\Rightarrow y = 2(x - 7)^2 - 4
$$

✔ Answer: $ y = 2(x - 7)^2 - 4 $

---

10) $ 6x^2 + 12x + y + 13 = 0 $

Solve for $y$:

$$
y = -6x^2 - 12x - 13
$$

Factor out $-6$ from first two terms:

$$
y = -6(x^2 + 2x) - 13
$$

- Half of 2: 1, square: 1
- Add and subtract 1:

$$
y = -6(x^2 + 2x + 1 - 1) - 13 = -6((x + 1)^2 - 1) - 13
$$

Distribute:

$$
y = -6(x + 1)^2 + 6 - 13 = -6(x + 1)^2 - 7
$$

✔ Answer: $ y = -6(x + 1)^2 - 7 $

---

11) $ 162x + 731 = -y - 9x^2 $

Solve for $y$:

$$
y = -162x - 731 - 9x^2
\Rightarrow y = -9x^2 - 162x - 731
$$

Factor out $-9$:

$$
y = -9(x^2 + 18x) - 731
$$

- Half of 18: 9, square: 81
- Add and subtract 81:

$$
y = -9(x^2 + 18x + 81 - 81) - 731 = -9((x + 9)^2 - 81) - 731
$$

Distribute:

$$
y = -9(x + 9)^2 + 729 - 731 = -9(x + 9)^2 - 2
$$

✔ Answer: $ y = -9(x + 9)^2 - 2 $

---

12) $ x^2 - 12x + y + 40 = 0 $

Solve for $y$:

$$
y = -x^2 + 12x - 40
$$

Factor out $-1$:

$$
y = -(x^2 - 12x) - 40
$$

- Half of $-12$: $-6$, square: 36
- Add and subtract 36:

$$
y = -(x^2 - 12x + 36 - 36) - 40 = -((x - 6)^2 - 36) - 40
$$

Distribute:

$$
y = -(x - 6)^2 + 36 - 40 = -(x - 6)^2 - 4
$$

✔ Answer: $ y = -(x - 6)^2 - 4 $

---

13) $ y = x^2 + 10x + 33 $

- Half of 10: 5, square: 25
- Add and subtract 25:

$$
y = x^2 + 10x + 25 - 25 + 33 = (x + 5)^2 + 8
$$

✔ Answer: $ y = (x + 5)^2 + 8 $

---

14) $ y + 6 = (x + 3)^2 $

Solve for $y$:

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

✔ Answer: $ y = (x + 3)^2 - 6 $

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✔ Final Answers:

| Problem | Vertex Form |
|--------|-------------|
| 1 | $ y = (x + 8)^2 + 7 $ |
| 2 | $ y = (x - 1)^2 - 6 $ |
| 3 | $ y = -(x + 7)^2 - 10 $ |
| 4 | $ y = 2(x + 9)^2 + 8 $ |
| 5 | $ y = (x - 6)^2 + 10 $ |
| 6 | $ y = (x + 2)^2 - 4 $ |
| 7 | $ y = (x - 3)^2 - 4 $ |
| 8 | $ y = \left(x + \frac{9}{2}\right)^2 - \frac{1}{4} $ |
| 9 | $ y = 2(x - 7)^2 - 4 $ |
|10 | $ y = -6(x + 1)^2 - 7 $ |
|11 | $ y = -9(x + 9)^2 - 2 $ |
|12 | $ y = -(x - 6)^2 - 4 $ |
|13 | $ y = (x + 5)^2 + 8 $ |
|14 | $ y = (x + 3)^2 - 6 $ |

Let me know if you'd like these graphed or explained further!