Quadratic Function Worksheets - Free Printable

It looks like you've uploaded a worksheet titled "Vertex Form" that asks students to rewrite quadratic equations in vertex form. The vertex form of a quadratic equation is:

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

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

Let’s go through each problem one by one and convert them into vertex form by completing the square.

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

We complete the square:
- Take coefficient of $x$: $-6$
- Half of it: $-3$
- Square it: $9$

Now:
$$
f(x) = x^2 - 6x + 9 - 9 + 5 = (x - 3)^2 - 4
$$

✔ Answer: $ f(x) = (x - 3)^2 - 4 $

---

2. $ f(x) = -3x^2 + 24x - 45 $

Factor out $-3$ from the first two terms:
$$
f(x) = -3(x^2 - 8x) - 45
$$

Complete the square inside the parentheses:
- Half of $-8$ is $-4$, square is $16$

Add and subtract $16$ inside the parentheses:
$$
f(x) = -3(x^2 - 8x + 16 - 16) - 45 = -3[(x - 4)^2 - 16] - 45
$$

Distribute:
$$
= -3(x - 4)^2 + 48 - 45 = -3(x - 4)^2 + 3
$$

✔ Answer: $ f(x) = -3(x - 4)^2 + 3 $

---

3. $ f(x) = 4x^2 + 12x - 10 $

Factor out 4:
$$
f(x) = 4(x^2 + 3x) - 10
$$

Complete the square:
- Half of 3 is $1.5$, square is $2.25$

$$
= 4(x^2 + 3x + 2.25 - 2.25) - 10 = 4[(x + 1.5)^2 - 2.25] - 10
$$

Distribute:
$$
= 4(x + 1.5)^2 - 9 - 10 = 4(x + 1.5)^2 - 19
$$

Note: $1.5 = \frac{3}{2}$, so we can write:
$$
f(x) = 4\left(x + \frac{3}{2}\right)^2 - 19
$$

✔ Answer: $ f(x) = 4\left(x + \frac{3}{2}\right)^2 - 19 $

---

4. $ f(x) = -4x^2 + 12x - 21 $

Factor out $-4$:
$$
f(x) = -4(x^2 - 3x) - 21
$$

Complete the square:
- Half of $-3$ is $-1.5$, square is $2.25$

$$
= -4(x^2 - 3x + 2.25 - 2.25) - 21 = -4[(x - 1.5)^2 - 2.25] - 21
$$

Distribute:
$$
= -4(x - 1.5)^2 + 9 - 21 = -4(x - 1.5)^2 - 12
$$

In fractions: $1.5 = \frac{3}{2}$, so:
$$
f(x) = -4\left(x - \frac{3}{2}\right)^2 - 12
$$

✔ Answer: $ f(x) = -4\left(x - \frac{3}{2}\right)^2 - 12 $

---

5. $ f(x) = 4x^2 + 8x + 7 $

Factor out 4:
$$
f(x) = 4(x^2 + 2x) + 7
$$

Complete the square:
- Half of 2 is 1, square is 1

$$
= 4(x^2 + 2x + 1 - 1) + 7 = 4[(x + 1)^2 - 1] + 7
$$

Distribute:
$$
= 4(x + 1)^2 - 4 + 7 = 4(x + 1)^2 + 3
$$

✔ Answer: $ f(x) = 4(x + 1)^2 + 3 $

---

6. $ f(x) = 5x^2 + 4x - 4 $

Factor out 5:
$$
f(x) = 5(x^2 + \frac{4}{5}x) - 4
$$

Complete the square:
- Half of $\frac{4}{5}$ is $\frac{2}{5}$, square is $\frac{4}{25}$

$$
= 5\left(x^2 + \frac{4}{5}x + \frac{4}{25} - \frac{4}{25}\right) - 4 = 5\left[\left(x + \frac{2}{5}\right)^2 - \frac{4}{25}\right] - 4
$$

Distribute:
$$
= 5\left(x + \frac{2}{5}\right)^2 - \frac{20}{25} - 4 = 5\left(x + \frac{2}{5}\right)^2 - \frac{4}{5} - 4
$$

Convert 4 to fifths: $4 = \frac{20}{5}$, so:
$$
= 5\left(x + \frac{2}{5}\right)^2 - \frac{24}{5}
$$

✔ Answer: $ f(x) = 5\left(x + \frac{2}{5}\right)^2 - \frac{24}{5} $

---

7. $ f(x) = 3x^2 + 24x - 20 $

Factor out 3:
$$
f(x) = 3(x^2 + 8x) - 20
$$

Complete the square:
- Half of 8 is 4, square is 16

$$
= 3(x^2 + 8x + 16 - 16) - 20 = 3[(x + 4)^2 - 16] - 20
$$

Distribute:
$$
= 3(x + 4)^2 - 48 - 20 = 3(x + 4)^2 - 68
$$

✔ Answer: $ f(x) = 3(x + 4)^2 - 68 $

---

8. $ f(x) = (2x - 3)(5x - 1) $

First expand:
$$
= 2x(5x - 1) - 3(5x - 1) = 10x^2 - 2x - 15x + 3 = 10x^2 - 17x + 3
$$

Now write in vertex form:
Factor out 10:
$$
f(x) = 10(x^2 - \frac{17}{10}x) + 3
$$

Complete the square:
- Half of $\frac{17}{10}$ is $\frac{17}{20}$, square is $\frac{289}{400}$

$$
= 10\left(x^2 - \frac{17}{10}x + \frac{289}{400} - \frac{289}{400}\right) + 3
= 10\left[\left(x - \frac{17}{20}\right)^2 - \frac{289}{400}\right] + 3
$$

Distribute:
$$
= 10\left(x - \frac{17}{20}\right)^2 - \frac{2890}{400} + 3
= 10\left(x - \frac{17}{20}\right)^2 - \frac{289}{40} + 3
$$

Convert 3 to fortieths: $3 = \frac{120}{40}$

$$
= 10\left(x - \frac{17}{20}\right)^2 - \frac{169}{40}
$$

✔ Answer: $ f(x) = 10\left(x - \frac{17}{20}\right)^2 - \frac{169}{40} $

---

9. $ f(x) = 4(2x - 7)(x + 1) $

First expand:
$$
= 4[(2x)(x + 1) - 7(x + 1)] = 4[2x^2 + 2x - 7x - 7] = 4[2x^2 - 5x - 7]
$$

Multiply:
$$
f(x) = 8x^2 - 20x - 28
$$

Now complete the square:
Factor out 8:
$$
f(x) = 8(x^2 - \frac{20}{8}x) - 28 = 8(x^2 - 2.5x) - 28
$$

Half of $2.5 = 1.25$, square is $1.5625 = \frac{25}{16}$

$$
= 8\left(x^2 - 2.5x + \frac{25}{16} - \frac{25}{16}\right) - 28
= 8\left[\left(x - \frac{5}{4}\right)^2 - \frac{25}{16}\right] - 28
$$

Distribute:
$$
= 8\left(x - \frac{5}{4}\right)^2 - \frac{200}{16} - 28 = 8\left(x - \frac{5}{4}\right)^2 - 12.5 - 28
= 8\left(x - \frac{5}{4}\right)^2 - 40.5
$$

As fraction: $40.5 = \frac{81}{2}$

So:
$$
f(x) = 8\left(x - \frac{5}{4}\right)^2 - \frac{81}{2}
$$

✔ Answer: $ f(x) = 8\left(x - \frac{5}{4}\right)^2 - \frac{81}{2} $

---

10. $ f(x) = 3x^2 + 6x + 30 $

Factor out 3:
$$
f(x) = 3(x^2 + 2x) + 30
$$

Complete the square:
- Half of 2 is 1, square is 1

$$
= 3(x^2 + 2x + 1 - 1) + 30 = 3[(x + 1)^2 - 1] + 30
= 3(x + 1)^2 - 3 + 30 = 3(x + 1)^2 + 27
$$

✔ Answer: $ f(x) = 3(x + 1)^2 + 27 $

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✔ Final Answers Summary:

| Problem | Vertex Form |
|--------|-------------|
| 1 | $ f(x) = (x - 3)^2 - 4 $ |
| 2 | $ f(x) = -3(x - 4)^2 + 3 $ |
| 3 | $ f(x) = 4\left(x + \frac{3}{2}\right)^2 - 19 $ |
| 4 | $ f(x) = -4\left(x - \frac{3}{2}\right)^2 - 12 $ |
| 5 | $ f(x) = 4(x + 1)^2 + 3 $ |
| 6 | $ f(x) = 5\left(x + \frac{2}{5}\right)^2 - \frac{24}{5} $ |
| 7 | $ f(x) = 3(x + 4)^2 - 68 $ |
| 8 | $ f(x) = 10\left(x - \frac{17}{20}\right)^2 - \frac{169}{40} $ |
| 9 | $ f(x) = 8\left(x - \frac{5}{4}\right)^2 - \frac{81}{2} $ |
| 10 | $ f(x) = 3(x + 1)^2 + 27 $ |

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🔍 How to Solve These:

To convert any quadratic to vertex form, follow these steps:

1. Factor out the leading coefficient (if not 1) from the $x^2$ and $x$ terms.
2. Complete the square inside the parentheses:
- Take half the coefficient of $x$, square it.
- Add and subtract this value inside the parentheses.
3. Rewrite as a perfect square trinomial.
4. Simplify constants outside.

This method works for all quadratics!

Let me know if you'd like a visual explanation or graphing tips!