Converting Quadratic Equations Worksheet - Coach Forrester - Free Printable

Problem Overview:

The task involves converting quadratic equations between vertex form and standard form. Let's break it down into two parts:

1. Converting from Vertex Form to Standard Form:
- Vertex form: \( y = a(x - h)^2 + k \)
- Standard form: \( y = ax^2 + bx + c \)

2. Converting from Standard Form to Vertex Form:
- Standard form: \( y = ax^2 + bx + c \)
- Vertex form: \( y = a(x - h)^2 + k \)

We will solve each problem step by step.

---

Part 1: Converting from Vertex Form to Standard Form

#### 1) \( y = -(x - 1)^2 - 1 \)
- Expand \( (x - 1)^2 \):
\[
(x - 1)^2 = x^2 - 2x + 1
\]
- Substitute back into the equation:
\[
y = -(x^2 - 2x + 1) - 1
\]
- Distribute the negative sign:
\[
y = -x^2 + 2x - 1 - 1
\]
- Simplify:
\[
y = -x^2 + 2x - 2
\]

Answer: \( y = -x^2 + 2x - 2 \)

#### 2) \( y = 2(x - 2)^2 - 3 \)
- Expand \( (x - 2)^2 \):
\[
(x - 2)^2 = x^2 - 4x + 4
\]
- Substitute back into the equation:
\[
y = 2(x^2 - 4x + 4) - 3
\]
- Distribute the 2:
\[
y = 2x^2 - 8x + 8 - 3
\]
- Simplify:
\[
y = 2x^2 - 8x + 5
\]

Answer: \( y = 2x^2 - 8x + 5 \)

#### 3) \( y = (x + 4)^2 + 4 \)
- Expand \( (x + 4)^2 \):
\[
(x + 4)^2 = x^2 + 8x + 16
\]
- Substitute back into the equation:
\[
y = x^2 + 8x + 16 + 4
\]
- Simplify:
\[
y = x^2 + 8x + 20
\]

Answer: \( y = x^2 + 8x + 20 \)

---

Part 2: Converting from Standard Form to Vertex Form

#### 4) \( y = x^2 - 8x + 15 \)
- To convert to vertex form, complete the square.
- Start with the quadratic and linear terms:
\[
y = x^2 - 8x + 15
\]
- Take the coefficient of \( x \), which is \(-8\), divide by 2, and square it:
\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\]
- Add and subtract 16 inside the equation:
\[
y = (x^2 - 8x + 16) - 16 + 15
\]
- Factor the perfect square trinomial:
\[
y = (x - 4)^2 - 1
\]

Answer: \( y = (x - 4)^2 - 1 \)

#### 5) \( y = x^2 - 4x \)
- Complete the square.
- Take the coefficient of \( x \), which is \(-4\), divide by 2, and square it:
\[
\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4
\]
- Add and subtract 4 inside the equation:
\[
y = (x^2 - 4x + 4) - 4
\]
- Factor the perfect square trinomial:
\[
y = (x - 2)^2 - 4
\]

Answer: \( y = (x - 2)^2 - 4 \)

#### 6) \( y = x^2 + 8x + 18 \)
- Complete the square.
- Take the coefficient of \( x \), which is \( 8 \), divide by 2, and square it:
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]
- Add and subtract 16 inside the equation:
\[
y = (x^2 + 8x + 16) - 16 + 18
\]
- Factor the perfect square trinomial:
\[
y = (x + 4)^2 + 2
\]

Answer: \( y = (x + 4)^2 + 2 \)

#### 7) \( y = x^2 + 4x + 3 \)
- Complete the square.
- Take the coefficient of \( x \), which is \( 4 \), divide by 2, and square it:
\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]
- Add and subtract 4 inside the equation:
\[
y = (x^2 + 4x + 4) - 4 + 3
\]
- Factor the perfect square trinomial:
\[
y = (x + 2)^2 - 1
\]

Answer: \( y = (x + 2)^2 - 1 \)

#### 8) \( y = x^2 - 2x + 5 \)
- Complete the square.
- Take the coefficient of \( x \), which is \(-2\), divide by 2, and square it:
\[
\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1
\]
- Add and subtract 1 inside the equation:
\[
y = (x^2 - 2x + 1) - 1 + 5
\]
- Factor the perfect square trinomial:
\[
y = (x - 1)^2 + 4
\]

Answer: \( y = (x - 1)^2 + 4 \)

#### 9) \( y = x^2 - 8x + 17 \)
- Complete the square.
- Take the coefficient of \( x \), which is \(-8\), divide by 2, and square it:
\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\]
- Add and subtract 16 inside the equation:
\[
y = (x^2 - 8x + 16) - 16 + 17
\]
- Factor the perfect square trinomial:
\[
y = (x - 4)^2 + 1
\]

Answer: \( y = (x - 4)^2 + 1 \)

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Final Answers:

1. \( y = -x^2 + 2x - 2 \)
2. \( y = 2x^2 - 8x + 5 \)
3. \( y = x^2 + 8x + 20 \)
4. \( y = (x - 4)^2 - 1 \)
5. \( y = (x - 2)^2 - 4 \)
6. \( y = (x + 4)^2 + 2 \)
7. \( y = (x + 2)^2 - 1 \)
8. \( y = (x - 1)^2 + 4 \)
9. \( y = (x - 4)^2 + 1 \)

\boxed{
\begin{aligned}
&1. \, y = -x^2 + 2x - 2 \\
&2. \, y = 2x^2 - 8x + 5 \\
&3. \, y = x^2 + 8x + 20 \\
&4. \, y = (x - 4)^2 - 1 \\
&5. \, y = (x - 2)^2 - 4 \\
&6. \, y = (x + 4)^2 + 2 \\
&7. \, y = (x + 2)^2 - 1 \\
&8. \, y = (x - 1)^2 + 4 \\
&9. \, y = (x - 4)^2 + 1
\end{aligned}
}