Quadratic Functions

by Shannon Jarret

1. Standard Form y=a(x-h)²+k

1.1. Graphing

1.1.1. domain and range

1.1.2. transformations

1.1.2.1. changes in a

1.1.2.2. changes in h

1.1.2.2.1. h>0, parabola shifts right

1.1.2.2.2. h<0, parabola shifts left

1.1.2.3. changes in k

1.1.2.3.1. k>0, parabola shifts up

1.1.2.3.2. k<0, parabola shifts down

1.1.3. function

1.1.3.1. Vertical line test

1.2. (h,k) = vertex

1.2.1. if a>0, vertex is the minimum

1.2.2. if a<0, vertex is the maximum

1.2.3. x=h is the axis of symmetry

2. General Form y=ax²+ bx + c

2.1. Graphing

2.1.1. domain and range

2.1.2. transformations

2.1.2.1. changes in a (vertical stretch)

2.1.2.1.1. if a>0, the parabola opens up

2.1.2.1.2. if a<0, the parabola opens down

2.1.2.1.3. if -1<a<1 and a≠0, the parabola widens

2.1.2.1.4. if a>1 or a<-1, the parabola narrows

2.1.2.2. changes in b

2.1.2.2.1. b<0, vertex moves down and right

2.1.2.2.2. b>0, vertex moves down and left

2.1.2.3. changes in c

2.1.2.3.1. c>0, parabola moves up

2.1.2.3.2. c<0, parabola moves down

2.1.3. function

2.1.3.1. vertical line test

2.2. Completing the square

2.3. Univariate polynomial equation with degree 2

2.4. c is the y-intercept

3. Quadratic formula

3.1. roots/zeros

3.2. discriminant

3.2.1. b²-4ac

3.2.1.1. if b²-4ac=0, we have one distinct root

3.2.1.2. if b²-4ac>0, we have two dinstinct roots

3.2.1.3. if b²-4ac<0, we have two imaginary roots

4. Quadratic Equation

4.1. set function equal to 0