Sketching Quadratic Functions

When sketching the curve of a quadratic function, we need only to represent critical information about the curve:

• The $y$ intercept, $(0,c)$ extracted from $ax^2+bx+c$
• The $x$ intercepts a.k.a. the roots, found by solving the quadratic equation for $x$ when $y=0$
• The turning point, $(-h,k)$ extracted from $a(x+h)^2+k$

We are not drawing so we do not need to plot every point.

The nature of the roots

We use the discriminant, $b^2-4ac$ to determine the nature of the roots:

• $b^2-4ac>0$: the curve will have $2$ real distinct roots. It will cut the $x$ axis at $2$ different points.
• $b^2-4ac=0$: the curve will have $2$ real equal/repeated roots. It will cut the $x$ axis twice at the same point (appearing to touch the axis at a single point).
• $b^2-4ac<0$: the curve will have $2$ complex roots. It will not cut the $x$ axis at all. This will appear as a smile that floats above the $x$ axis for minimum curves or a frown that floats below the $x$ axis for maximum curves.