# Graphing

## Parabolas

A parabola is the graph of a quadratic polynomial in one variable (see more in the Polynomials section). Its general equation comes in three forms: \begin{array}{l l} \text{Standard form: } & y = ax^2 + bx + c \\ \text{Vertex form: } & y = a(x-h)^2 + k \\ \text{Factored form: } & y = a(x-r)(x-s) \end{array} The factored form of the equation tells us the roots, i.e. the $x$-intercepts, $x=r$ and $x=s$.

The key information in drawing a parabola is the vertex, which we can read off from the vertex form equation as the point $(h,k)$.

If $a>0$, the parabola opens upwards. Otherwise, for $a<0$, it opens downwards as illustrated below.

 A parabola in the case $a<0$ A parabola in the case $a>0$

Note: For larger $|a|$, the graph will be narrower and for smaller $|a|$ the graph will be wider.

The graph of $y=ax^2$ illustrated for $a>1$ (red), $a=1$ (blue), and $0 < a < 1$ (green)

Drawing the parabola is easier if we have the vertex form of the equation, so we need to know how to go from the standard to the vertex form.

### Completing the square

Going from the standard form to the vertex form of a quadratic equation involves "completing the square", as illustrated below: \begin{align*} y &= ax^2 + bx + c \\ y &= a\left(x^2 + \frac{b}{a}x\right) + c \\ y &= a\left(x^2+\frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \right) + c \\ y &= a\left(x^2+\frac{b}{a}x + \frac{b^2}{4a^2}\right) - a \left(\frac{b^2}{4a^2}\right) + c \\ y &= a\left(x + \frac{b}{2a} \right)^2 - \left(\frac{b^2}{4a} - c \right) \\ y &= a\left(x + \frac{b}{2a} \right)^2 - \frac{b^2-4ac}{4a} \end{align*}

### Example. Draw the parabola with equation $y = 3x^2 + 12x + 8$.

Solution: Completing the square, we get that the vertex form of the equation of this parabola is: $$y = 3(x + 2)^2 - 4$$ So, the vertex of this parabola is $(-2,-4)$.

It intersects the $y$-axis when $x=0$, then we get $y = 3(2)^2-4=8$. It intersects the $x$-axis when $y=0$, then $0 = 3(x+2)^2 - 4$ or $(x+2)^2=4/3$ or $x=-2 \pm \sqrt{4/3}$.

Given this information, we can draw the parabola:

The parabola $y = 3(x + 2)^2 - 4$