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!