Quadratics - Michael Chen

A quadratic equation or function can be written in Standard, Factored or Vertex form.

Equation

$y=ax^2+bx+c$
$f(x)=ax^2+bx+c$

$y=a(x−r_1)(x−r_2)$
$y=a(x−r_0)^2$
$f(x)=a(x−r_1)(x−r_2)$
$f(x)=a(x−r_0)^2$

$y=a(x−h)^2+k$
$f(x)=a(x−h)^2+k$

Opens up or down

If $a > 0$, parabola opens up (smiley).
If $a < 0$, parabola opens down (sad).

x-intercepts

$x={-b±√{b^2-4ac}}/{2a}$

$x=r_1$ and $x=r_2$ (two intercepts)

or

$x=r_0$ (one intercept)

Solve for $x$ by setting $y=0$.

(Having a solution means one or two intercepts. No solution means no intercept.)

y-intercept

$c$

Solve for $y$ by setting $x=0$.

Axis (line) of symmetry

$x=-b/{2a}$

$x={r_1+r_2}/2$

$x=h$

Vertex

Calculate $x=-b/{2a}$. Substitute into equation to get y. Vertex is (x,y).

Calculate $x={r_1+r_2}/2$. Substitute into equation to get y. Vertex is (x,y).

$(h, k)$

Minimum / Maximum

Calculate the vertex.
If $a > 0$, $y$ is the minimum.
If $a < 0$, $y$ is the maximum.

If $a > 0$, $k$ is the minimum.
If $a < 0$, $k$ is the maximum.

Domain

TBD

Range

TBD

X-intercepts

There can be zero, one or two x-intercepts. It is easiest to tell in Standard Form, $y=ax^2+bx+c$.

FYI, the $b^2-4ac$ part is called the discriminant.