Quadratics Functions

Forms of
Quadratic Equations

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

Vertex/Axis Of Symmetry (AOS)
= (h,k)

h is the x value of
vertex and AOS

k is the y value of
vertex and AOS

Transformation
--------------------------
y= (x - h)^2, then the
parabola is shifted
right by h units.

y= (x+ h)^2, then the
parabola is shifted left
by h units

y= x^2 - k, then the
parabola is shifted
down by k units.

y= x^2 + k, then the
parabola is shifted up
by k units.

Ming throws a stone off a bridge into a river below. The stone's height (in meters above the water), x seconds after Ming threw it, it is modeled by:
h(x)=-5(x-1)^2+45

What is the maximum height that the stone will reach?

The maximum height is reached at the vertex. So, in order to find the maximum height, we need to find the vertex's y-coordinate. The vertex is (1, 45). So, in conclusion, the stone reaches the maximum height at 45 meters.

Standard Form:
y=ax^2+bx+c

c value is the
y-intercept
of the parabola

a is the stretch/
compression factor

Direction of Opening
-------------------------
If a>0 it is opening
upwards. If a<0 it is
opening downwards.

Transformation of Parabola
--------------------------------
If a>1 then the graph is
stretching vertically by a
value. If 0<a<1 (a is
a fraction) then the graph
is compressed vertically.
y=ax^2, the parabola is
reflected over the x-axis.

Step Pattern
---------------------
Multiply a value
and step pattern
to get the correct
points of the
parabola

(1,3,5) * a

Formula for AOS
x = -b/2a

a,b, and c are real
numbers and a is
not equal to 0

A rock is thrown from the top of a tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by d(t) = -16t^2 – 4t + 382. How long after the rock is thrown is it 370 feet from the ground?
---------------------------------------------------
d(t) = 16t^2 - 4t +382
-16t^2 - 4t + 382 = 370
-16t^2 - 4t + 382 - 370 = 0
-16t^2 - 4t + 12 = 0
16t^2 + 4t - 12 = 0
4t^2 + t - 3 = 0
(4t - 3)(t + 1) = 0
4t - 3 = 0 or t + 1 = 0
t = 3/4 or t = -1

Therefore, it takes 75 seconds to reach 370 feet.

Factored Form:
y=a(x-r)(x-s)

Binomials (x-r) and
(x-s) gives the x -
intercepts.

y= 0.5(x-6)(x+2)
--------------------
(x-6)= 0 (x+2)=0
x=0+6 x=0-2
x= 6 x= -2

To find the x-value
of the vertex, use the
formula (r+s)/2

Solving Quadratic
Equations

Factor

Common Factor

Find the GCF and take
it out by dividing each
term by the GCF.

6x^2 - 2x = 0
2x(3x-1)=0
2x=0 3x-1=0
x=0 x=1/3

Factor by Grouping
-----------------------
1) group terms with
like terms
2)factor each group to
get a binomial common
factor

3x^2+6x+4x+8
=(3x^2+6x)+(4x+8)
= 3x(x+2)+4(x+2)
= (3x+4)(x+2)

Simple Trinomial

Find 2 numbers that
multiply to c and add
to b.

1) Use Criss-Cross Method
2) MAN Method
(Multiply, Add, Number)

x^2-2x-15
M- -15
A- -2
N- (-5,3)

=(x-5)(x+3)

Complex Trinomial

Perfect Square
Trinomial
-----------------------
(a+b)^2
= a^2 + 2ab + b^2

(a-b)^2
= a^2 - 2ab - b^2

(3x+2y)^2
=3^2 + 2(3)(2) + (2)^2
= 9x^2+12xy+4y^2

Difference of Squares
--------------------------
(a+b)(a-b)
= a^2 - b^2

(9x^2-16)
=(3x)^2 - 4^2
= (3x+4)(3x-4)

Expanding

FOIL Method
---------------
First Outer
Inner Last

(x+5)(x+2)
=x^2+2x+5x+10
=x^2+7x+10

Complete the
Square

Convert Standard Form
into Vertex Form and
use this to find min./max.
value or vertex of parabola.

1) Start by factoring out the a
2) Move the c term to the other side of the equation.
3) Use the b term in order to find a new c term that makes a perfect square. This is done by first dividing the b term by 2 and squaring the quotient and add to both sides of the equation.
4) Find your h, the b term divided by two, for the perfect square.
5) Set equation to zero.

x^2+2x-8=0
(x+1)^2 - (1)^2 - 8 =0
(x+1)^2-1-8=0
(x+1)^2-9=0
(x+1)^2=9
x+1=+- (square root of 9)
x=-3-1 x=3-1
x=-4 x=2

Using Quadratic
Formula

Discriminant
-------------
b^2 - 4ac

1) If the discriminant is
greater than 0, then there
are 2 real roots

2)If the discriminant is
equal to 0, then there
is 1 real root.

3) If the discriminant is
than 0, then there are
no real roots.

Parts of Parabolas
and Definitions

Y-Intercept
--------------
The coordinate
where the parabola
crosses the y-axis.

Optimal Value
----------------
The y-value of
the vertex.

Vertex
---------------------
A parabola
has a minimum
(opens upward)
& maximum (opens
downward) value.
It is also the point
of AOS.

Axis of Symmetry (AOS)
----------------------------
A vertical line that depicts
the point of symmetry

Zeros/Roots
---------------
When the parabola
crosses the x-axis,
the x-coordinate it is
called zeros or the
x-intercept.