Question

How do you graph a quadratic function?

Answer 1

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=ax^2+bx+c` look like `y=a(x-h)^2+k`, then the parabola's vertex (tip) is at `(h,k).`

Example:
Say your equation is given as: `y=x^2-6x+8.` Here `a=1`.
Notice that `(x-h)^2` has a middle term of `-2hx`,
so we need to make `h = 6/3 = 2.`

So `(x-3)^2 = x^2 - 6x + 9,` but we want a constant term of 8, meaning `y=(x-3)^2-1.`

We read the vertex at `(3,-1)` and the y-intercept at `(0,8)`.
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 `(2,0)` and `(4,0).` (How do these relate to the factors of `x^2-6x+8` ?)

You're welcome, and happy graphing from `dansmath.`