# How do you graph a quadratic function?

Oct 10, 2014

My favorite way is to complete the square in the equation for y, find the vertex and y-intercept, and draw the parabola.
Our goal is to make the equation $y = a {x}^{2} + b x + c$ look like $y = a {\left(x - h\right)}^{2} + k$, then the parabola's vertex (tip) is at $\left(h , k\right) .$

Example:
Say your equation is given as: $y = {x}^{2} - 6 x + 8.$ Here $a = 1$.
Notice that ${\left(x - h\right)}^{2}$ has a middle term of $- 2 h x$,
so we need to make $h = \frac{6}{3} = 2.$

So ${\left(x - 3\right)}^{2} = {x}^{2} - 6 x + 9 ,$ but we want a constant term of 8, meaning $y = {\left(x - 3\right)}^{2} - 1.$

We read the vertex at $\left(3 , - 1\right)$ and the y-intercept at $\left(0 , 8\right)$.
This gives a parabola sitting on the vertex and curving upward, with axis of symmetry on the vertical line $x = 3.$

By the way this example also has roots (x-intercepts) at $\left(2 , 0\right)$ and $\left(4 , 0\right) .$ (How do these relate to the factors of ${x}^{2} - 6 x + 8$ ?)

You're welcome, and happy graphing from $\mathrm{da} n s m a t h .$