Quadratic Functions

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


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.

Drag the blue point.

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.

Change the value of \(a\).
What point is always on the graph regardless of the value of \(a\)?

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

\[ \binom{h}{k}. \]
Try to enter another function.

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 \(f(x) = 0\).

Change the value of \(a\).

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