Question

Determine the vertex form and the maximum or minimum value of the function.
f (x) = 2x=+8x+3

f(x) = 2(x+ 4)? - 13, the maximum value of fis - 13
f(x) = 2(x+4) - 13, the minimum value of fis - 13
f(x) = 2 (x+2) - 5, the maximum value of fis -5
f(x) = 2(x+2)} - 5, the minimum value of fis - 5

Answer 1

Let's start with the given function:

\[
f(x) = 2x^2 + 8x + 3
\]

To determine the vertex form of the quadratic function, we will complete the square.

1. **Factor out the coefficient of \(x^2\)** from the first two terms:

\[
f(x) = 2(x^2 + 4x) + 3
\]

2. **Complete the square** for \(x^2 + 4x\):

- Take half of the coefficient of \(x\), which is \(4\), to get \(2\), and square it to get \(4\).
- Add and subtract this square inside the parentheses:

\[
f(x) = 2(x^2 + 4x + 4 - 4) + 3
\]
\[
f(x) = 2((x + 2)^2 - 4) + 3
\]
\[
f(x) = 2(x + 2)^2 - 8 + 3
\]
\[
f(x) = 2(x + 2)^2 - 5
\]

The vertex form of the function is:

\[
f(x) = 2(x + 2)^2 - 5
\]

3. **Determine the minimum or maximum value**:

Since the coefficient of \((x + 2)^2\) is \(2\) (which is positive), this means the parabola opens upwards, and there is a **minimum** point at the vertex.

4. **Minimum value**:

The minimum value of \(f(x)\) occurs at the vertex, given by the constant term:

\[
\text{Minimum value} = -5
\]

So, the vertex form of the function is:

\[
f(x) = 2(x + 2)^2 - 5
\]

And the minimum value of \(f\) is \(-5\).

### Summary:
The correct form is:
- \(f(x) = 2(x + 2)^2 - 5\), with the **minimum value** of \(f\) being \(-5\).