how do you put f (x) = 2x^2 - 6x +7 into vertex form

To convert the quadratic function \( f(x) = 2x^2 - 6x + 7 \) into **vertex form**, we follow the method of **completing the square**. The vertex form of a quadratic function is given by:

\[
f(x) = a(x-h)^2 + k
\]

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

### Steps:
1. **Factor out the coefficient of \(x^2\)** from the first two terms:
   \[
   f(x) = 2(x^2 - 3x) + 7
   \]

2. **Complete the square** inside the parentheses:
   - Take the coefficient of \(x\) (which is \(-3\)), divide it by 2, and square it:
     \[
     \left(\frac{-3}{2}\right)^2 = \frac{9}{4}
     \]
   - Add and subtract this value inside the parentheses:
     \[
     f(x) = 2\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) + 7
     \]
   - Group the perfect square trinomial:
     \[
     f(x) = 2\left(\left(x - \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 7
     \]

3. **Simplify** the equation:
   - Distribute the 2:
     \[
     f(x) = 2\left(x - \frac{3}{2}\right)^2 - 2 \times \frac{9}{4} + 7
     \]
     \[
     f(x) = 2\left(x - \frac{3}{2}\right)^2 - \frac{9}{2} + 7
     \]

   - Combine constants:
     \[
     f(x) = 2\left(x - \frac{3}{2}\right)^2 + \frac{5}{2}
     \]

### Final Answer:
The vertex form of the quadratic function is:

\[
f(x) = 2\left(x - \frac{3}{2}\right)^2 + \frac{5}{2}
\]

In this form, the vertex is \(\left(\frac{3}{2}, \frac{5}{2}\right)\).

Would you like a more detailed explanation of any step?

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Here are 5 related questions to explore further:
1. How can you find the vertex directly from the standard form?
2. What is the significance of the vertex in a quadratic function?
3. How does completing the square help in solving quadratic equations?
4. How do transformations affect the graph of a quadratic function?
5. How can the quadratic formula be used to find the roots of \( f(x) \)?

**Tip:** The \( h \)-value in the vertex form \( (x-h)^2 \) shifts the parabola horizontally, while \( k \) shifts it vertically!

Learn how to convert the quadratic function f(x) = 2x^2 - 6x + 7 into vertex form using the method of completing the square. This step-by-step guide explains key algebraic concepts, including the vertex form equation and completing the square, suitable for Grades 9-12 students.

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