# Quadratic Functions

The most common way to write a quadratic function is to use **general form**:

\[f(x)=ax^2+bx+c\]

When analyzing
the graph of a quadratic function, or the correspondence between the graph and solutions to quadratic
equations, two other forms are more suitable: **vertex form**
and **factor form**.

In order to rewrite a quadratic function from general form to vertex form, you must know how to complete the square.

## Completing the square

Completing the square is a method used for solving quadratic equations and for rewriting quadratic functions to vertex form.

It is easy to solve a quadratic equation of the form \(x^2=c\), you just take the square root of both sides. You can also easily solve equations of the form \((ax+b)^2=c\), first take the square root of both sides, then solve for \(x\). When completing the square, you rewrite a quadratic equation to the form \((ax+b)^2=c\).

Let's say we want to solve an equation of the form

\[x^2+bx = a\]where both \(a\) and \(b\) are positive numbers. In this case, we can use areas to explain how to complete the square.

From the worksheet we get that

\[ \begin{align} x^2+bx &= a \\ x^2+bx + \left(\frac{b}{2}\right)^2 &= a + \left(\frac{b}{2}\right)^2 \\ \left(x+\frac{b}{2}\right)^2 &= a + \left(\frac{b}{2}\right)^2 \end{align} \]which is an equation we can solve.

It is possible to make a similar geometric construction for negative values of \(b\), but as long as we use a geometrical interpretation of the problem, \(a\) must not be negative and the geometrical solution will not yield negative \(x\)-values. In order to get all roots in the general case, it is better to use an algebraic approach.

The first person to find a method for solving quadratic functions was Brahmagupta, who also made one of the greatest contributions to mankind ever ‐ he showed how to do arithmetic calculations using the number zero.

## Vertex form

A quadratic function can always be written as

\[ f(x) = a(x-h)^2+k\]which is called **vertex form**.

### Translations of graphs

The graph of a quadratic function \(g(x)=(x-h)^2+k\), can be seen as the translation of the graph of the function \(f(x)=x^2\) along the vector

\[ \binom{h}{k}. \]### From general form to vertex form

To rewrite a quadratic function from general to vertex form, you can complete the square. In the general we get this:

\[ \begin{align} f(x) &= ax^2+bx+c \\ &= a\left( x^2+\frac{b}{a}x+\frac{c}{a}\right) \\ &= a\left( x^2+2\frac{b}{2a}x+ \left(\frac{b}{2a} \right)^2-\left(\frac{b}{2a} \right)^2+\frac{c}{a}\right) \\ &= a\left( \left(x + \frac{b}{2a}\right)^2 -\left(\frac{b}{2a} \right)^2+\frac{c}{a}\right) \\ &= a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a} \end{align} \]## Factor form

A quadratic function can always be written as

\[ f(x) = a(x-\alpha)(x-\beta\]which is called **vertex form**. The coefficients \(\alpha\) and \(\beta\) are roots to the quadratic equation \(f(x) = 0\).

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License