Topic 8: Parabolas

Sketching parabolas with various methods.

Sketching using Transformations

y=a(x-h)²+k

tp = (h,k), axis of symmetry at x = h

If a>0, the parabola is upright; a<0, inverted.

Sketching using Factorisation

To sketch a graph of a monic equation y=x²+bx+c:

Find y-int by sub x=0; find x-int by sub y=0 and factorise using the Null Factor Law.

Note: Null Factor Law: if a*b=0 then a=0/b=0

The tp is the axis of symmetry; use two x-values/int, sum, and divide by two.

Sub this x-value to find y coordinate of tp.

Sketching by completing the square

To sketch a quadratic in y=a(x-h)²+k (from ax²+bx+c):

Determine the tp (h,k): if a is positive, tp is min; if a is negative, tp is max.

Determine y-int by x=0; (if any) determine x-int by y=0.

Tip: to solve x²=a, a>0, take √a for roots.

Sketching using Quadratic Formula

For y=ax²+bx+c:

y-int at y=c (x=0); x-int at y=0;

Use quadratic formula for x-int:

Turning points: x=-b/2a (also axis of symmetry), y=c-b²/4a (or sub x value to original equation)

Discriminant (Δ): b²-4ac.

If Δ < 0, no real solutions (√-num =?)

If Δ = 0, one real solution (x=-b/2a, the only x intercept is the tp)

If Δ > 0, two real solutions (quadratic formula)