Question

f(x)= 8x2 + 8x- 12 . How do you convert in vertex form

Answer 1

Conditions

$f(x) = 8x^2 + 8x - 12$. How do you convert in vertex form

Solution

A quadratic function can be expressed in three formats:

- $f(x) = ax^2 + bx + c$ is called the standard form,

- $f(x) = a(x - x_1)(x - x_2)$ is called the factored form, where $x_1$ and $x_2$ are the roots of the quadratic equation, it is used in logistic map

- $f(x) = a(x - h)^2 + k$ is called the vertex form, where $h$ and $k$ are the $x$ and $y$ coordinates of the vertex, respectively.

To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots $x_1$ and $x_2$. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

Let's complete the square:

For this let's find the coordinates of the vertex. The vertex of the parabola in the vertex form is

$$\left(- \frac {b}{2 a}, - \frac {\Delta}{4 a}\right).$$

So,

$$f (x) = a \left(x - \frac {8}{1 6}\right) ^ {2} - \frac {6 4 + 4 \cdot 8 \cdot 1 2}{4 \cdot 8} = a \left(x - \frac {1}{2}\right) ^ {2} - 1 4$$

It's obvious, that $a$ is 8 here:

$$8 \left(x - \frac {1}{2}\right) ^ {2} - 1 4 = 8 \left(x ^ {2} - x + \frac {1}{4}\right) - 1 4 = 8 x ^ {2} - 8 x + 2 - 1 4 = f (x)$$

Answer: The vertex form:

$$f (x) = a \left(x - \frac {1}{2}\right) ^ {2} - 1 4$$