Question

This typically involves rewriting a quadratic function in the form
𝑦
=
𝑎
(
𝑥
−
ℎ
)
2
+
𝑘
, where
(
ℎ
,
𝑘
)
is the vertex of the parabola.

Answer 1

To rewrite a quadratic function in the vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex, you can follow these steps:

1. **Start with the standard form of a quadratic**: The standard form of a quadratic function is:
\[
y = ax^2 + bx + c
\]

2. **Factor out the coefficient of \(x^2\)**: If \(a \neq 1\), factor \(a\) out from the first two terms:
\[
y = a(x^2 + \frac{b}{a}x) + c
\]

3. **Complete the square**: Take the coefficient of \(x\) (which is \(\frac{b}{a}\)), divide it by 2, and square it. This value will help to complete the square.
- Let \(m = \frac{b}{2a}\), then \(m^2 = \left(\frac{b}{2a}\right)^2\).
- Add and subtract \(m^2\) inside the parentheses:
\[
y = a\left(x^2 + \frac{b}{a}x + m^2 - m^2\right) + c
\]

4. **Reorganize the equation**: Now you can rewrite the expression:
\[
y = a\left((x + m)^2 - m^2\right) + c
\]
Expand this:
\[
y = a(x + m)^2 - am^2 + c
\]

5. **Identify \(h\) and \(k\)**: From the equation above, compare it to the vertex form \(y = a(x - h)^2 + k\) to identify:
- \(h = -m = -\frac{b}{2a}\)
- \(k = c - am^2 = c - a\left(\frac{b}{2a}\right)^2\)

6. **Final vertex form**: Substitute \(h\) and \(k\) back into the vertex form:
\[
y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - a\left(\frac{b}{2a}\right)^2\right)
\]

### Example:

Let's apply these steps to the quadratic function \(y = 2x^2 + 8x + 5\).

1. **Start with the function**:
\[
y = 2x^2 + 8x + 5
\]

2. **Factor out the coefficient of \(x^2\)**:
\[
y = 2(x^2 + 4x) + 5
\]

3. **Complete the square**:
- The coefficient of \(x\) is \(4\).
- Half of \(4\) is \(2\), and \(2^2 = 4\).
\[
y = 2(x^2 + 4x + 4 - 4) + 5
\]
\[
y = 2((x + 2)^2 - 4) + 5
\]
\[
y = 2(x + 2)^2 - 8 + 5
\]

4. **Final form**:
\[
y = 2(x + 2)^2 - 3
\]

5. **Identify \(h\) and \(k\)**:
- Here \(h = -2\) and \(k = -3\).
- The vertex of the parabola is \((-2, -3)\).

This gives you the quadratic in vertex form:
\[
y = 2(x + 2)^2 - 3
\]