Reading a parabola at a glance

A parabola gives away almost everything about itself in five numbers. Learn to read them and you can sketch one without plotting a single point.

Take a concrete one

Let $f(x) = 2x^2 - 8x + 6$. Here $a = 2$, $b = -8$, $c = 6$.

Opening and width

Since $a = 2 > 0$ the parabola opens upward, and since $|a| > 1$ it is narrower than $y = x^2$.

Axis and vertex

The axis of symmetry is the vertical line

$$x = -\frac{b}{2a} = -\frac{-8}{4} = 2$$

and the vertex lies on it: $f(2) = 8 - 16 + 6 = -2$. So the vertex is $(2, -2)$.

Roots

Divide by $2$: $x^2 - 4x + 3 = 0$, which factors as $(x-1)(x-3)$. The roots are $x = 1$ and $x = 3$, symmetric about the axis $x = 2$, as they must be.

The y-intercept

Set $x = 0$: $f(0) = c = 6$.

All five at a glance

What	Formula	This parabola
Direction	sign of $a$	up
Axis	$x = -b/2a$	$x = 2$
Vertex	$(-b/2a,\ f(-b/2a))$	$(2, -2)$
Roots	$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$	$1$ and $3$
y-intercept	$c$	$6$

Try the same reading on every example in the Quadratic Equations explainer: the graph updates as you step through.