Quadratic Functions

The graph of a quadratic function is always a parabola. There are two main forms for quadratic functions you should be aware of for your calculus course.

1. Standard Form: $f(x) = ax^{2}+bx+c$ where $a \neq 0$
   • if $a>0$, the parabola opens upwards
   • if $a<0$ the parabola opens downwards
   • This form is useful when you need to factor or use the quadratic formula.
2. Vertex Form: $f(x) = a(x-p)^{2} + q$ where $a \neq 0$
   • if $a>0$, the parabola opens upwards
   • if $a<0$ the parabola opens downwards
   • $(p,q)$ is the vertex of the parabola
   • This form is useful when you need to sketch the function or find the vertex of the parabola.

Completing the Square

Given a quadratic function in standard form $f(x) = ax^{2}+bx+c$, we can find the vertex form by completing the square. Follow the steps below to complete the square.

1. If $a=1$ (i.e. $f(x) = x^{2}+bx+c$):
   1. Calculate $\displaystyle \left( \frac{b}{2} \right)^{2}$.
   2. Add and subtract this value to obtain $f(x) = \displaystyle x^{2}+bx +\left( \frac{b}{2} \right)^{2} - \left( \frac{b}{2} \right)^{2}+c$.
   3. The function simplifies to $\displaystyle f(x) = \left( x + \frac{b}{2} \right)^{2} - \frac{b^{2}}{4} +c$.
2. If $a \neq 1$:
   1. Factor out the coefficient $a$ from the first two terms: $\displaystyle f(x) = a \left( x^{2} + \frac{b}{a}x \right) + c$.
   2. Calculate $\displaystyle \left( \frac{b}{2a} \right)^{2}$.
   3. Add and subtract this value inside the parentheses to obtain $\displaystyle f(x) = a \left( x^{2} + \frac{b}{a}x +\left( \frac{b}{2a} \right)^{2} - \left( \frac{b}{2a} \right)^{2}\right) + c$.
   4. Remove $\displaystyle - \left( \frac{b}{2a} \right)^{2}$ from the parentheses by multiplying by $a$: $\displaystyle f(x) = a \left( x^{2} + \frac{b}{a}x +\left( \frac{b}{2a} \right)^{2} \right) - a\cdot \left( \frac{b}{2a} \right)^{2}+ c$.
   5. The function simplifies to $\displaystyle f(x) = a\left( x + \frac{b}{2a} \right)^{2} - \frac{b^{2}}{4a}+c$.

Basic Function Transformations

Below is a chart summarizing transformations of the basic quadratic function $f(x) = x^{2}$.

Form	Description
$f(x) + c = x^{2} + c$	Shift up ($c>0$) or down ($c< 0$) by $c$ units
$f(x+c) = (x+c)^{2}$	Shift left ($c>0$) or right ($c< 0$) by $c$ units
$-f(x) = -x^{2}$	Reflection over the $x$-axis
$f(-x) = (-x)^{2}$	Reflection over the $y$-axis
$cf(x) = cx^{2}$	Stretch ($c>1$) or compression ($0< c < 1$) by a factor of $c$ in the $y$-axis
$f(cx) = (cx)^{2}$	Stretch ($0< c < 1$) or compression ($c>1$) by a factor of $c$ in the $x$-axis