Quadratic Functions

Simplified Form

y=x^2

3 variations

Standard Form

y=ax^2+bx+c

y=x^2+3x-2

Alain throws a stone off a bridge into a river below.
The stone's height (in meters above the water), xx seconds after Alain threw it, is modeled by:
h(x)=-5x^2+10x+15
What is the height of the stone at the time it is thrown?

Vertex Form

y=a(x-h)^2+k

y=2(x+4)^2+2

y=(x-2)^2

A football is kicked into the air. Its height in meters after t seconds is given by h=-4.9(t-2.4)^2+29
What was the maximum height of the ball?
How high was the ball after 2s?Was the ball still in the air after 5s?
What was the distance of the stone after it reaches it's max height

variable h represents horizontal translations

positive

parabola moves left h amount

negative

parabola moves right h amount

variable k represents vertical translations

positive

parabola moves up k amount

negative

parabola moves down k amount

Factored Form

y=a(x-s)(x-r)

y=3(x-3)(x+2)

y=3(2x-3)(4x+4)

The company's annual profit (in millions of dollars) as a function of the app's price (in dollars) is modeled by
P=-2(x-3)(x-11)
Which app prices will result in \$0 annual profit?

Variables

Real Roots/x-intercepts/zeroes

Quadratic Formula

x = -b ± √b^2 - 4ac
________________
2a

solvable

answer is x-intercept(s)

used if factored form does not work

discriminant is negative number

equation has no solution

Discriminant

b^2-4ac

Nature of roots

b^2-4ac = 0

equation has 1 solution

b^2-4ac > 0

equation has 2 solutions (real roots)

A boat can cover 10 km up the stream and 5 km down the stream in 6 hours. If the speed of the stream is 1.5 km/h, find the speed of the boat in still water.

b^2-4ac < 0

equation has no solution

Perfect Square

Equation can be factorized to factored form

Factored form

(x-s) and (x-t)

(x-s)=0

X-value is the first x-intercept

If (x-s) is (x-3)

(x-3)=0

x-3=0
x=3

One x-intercept is (3,0)

(x-t)=0

X-value is the second x-intercept

If (x-t) is (2x+3)

(2x+3)=0

2x+3=0
2x=-3
x=-3/2
x=-1.5

One x-intercept is (-3/2,0) or (-1.5,0)

Vertex

Vertex Form

V(-h,k)

-h is x-coordinate of vertex

k is y-coordinate of vertex

V (-b/2a, b^2/4a - c)

-b/2a finds x-coordinate of vertex

b^2/4a - c finds y-coordinate of vertex

Roots

root(1) + root(2) /2 is x-coordinate of vertex

y-coordinate can be found

substitution

subbing x-coordinate of vertex into any quadratic equation finds y-coordinate of vertex

Axis of symmetry

x-value of vertex

Max/Min value

y-value of vertex

a value

Direction of opening

negative

parabola opens down

Reflection over x-axis

positive

parabola opens up

Vertical stretch/compression

a value is stretch/compression factor

1<a>0 or -1>a<0

vertical compression

a>1 or a<-1

vertical stretch

y-intercept

substituting x as 0 in quadratic equations

y=(x-1)^2
y=(0-1)^2
y=(-1)^2
y=1

Step Pattern

plot points

symmetrical curved and open plane(U-shaped)

Parabola

y-intercept

Co-ordinate where parabola meets y-axis

vertex

axis of symmetry meets parabola

x co-ordinate

axis of symmetry

vertical line that divides graph into 2 equal halves

sum of the roots divided by 2

y co-ordinate

max/min value

highest/lowest point on graph

parabola opens up

minimum value

parabola opens down

Maximum value

zeroes

co-ordinate(s) where parabola meets x-axis

0,1 or 2 zeroes

plotting vertex

plot next points by moving right/left 1 and up each consecutive step pattern amount and connecting

If vertex is (0,0), and the a value is 2
the step pattern is 2,6,10...

point on graph is mirrored

plotting 2 known/solvable points on both sides of parabola

Connecting points

1a,3a,5a,7a...

turns standard form to factored form

divide entire trinomial by common factor, monomial or binomial(if possible)

lal≠1

Find Two integers that multiply to "a(c)" and add to "b"

replace these 2 integers with b value

factor by grouping

2x^2 + 11x + 12
= 2x^2 + 8x + 3x + 12
= 2x(x + 4) + 3(x + 4)
= (2x + 3)(x + 4)

2x^2 + 8x + 3x + 12
= 2x(x + 4) + 3(x + 4)
= (2x + 3)(x + 4)

Complex Trinomial

lal=1

Find Two integers that multiply to "c" and add to "b"

These 2 numbers are the variables r and s in (x-s) and (x-r) with the original GCF being the a value

x^2 + 8x + 12
= x^2 + 6x + 2x + 12
= (x+2)(x+6)

possibly not factorizable

turn standard form to vertex form

completing the square

perfect square trinomial

the left side of standard form converts perfect square trinomial

y = 2x^2 - 16x - 1
y = (2x 2 - 16x) - 1
y = 2(x 2 - 8x) - 1
y = 2(x 2  - 8x + 16 - 16) - 1
y = 2(x 2 - 8x + 16) - 16(2) - 1
y = 2(x - 4)^ 2 - 33

Formulas

difference of squares

(a + b)(a – b) = a 2 – b 2

Distributive property

a(b + c) = ab + ac

perfect square trinomial

(a±b)^2=a^2±2ab+b^2

Table of values

First Differences

the subtraction of consecutive y-values

first differences are constant, there is linear relation(line)

Second differences

the subtraction of first differences

second differences are constant and not 0, there is quadratic relation(parabola)

neither difference is constant there is no relation