Quadratic Functions

Forms of Quadratic Equations

What are Quadratics and Properties of Parabolas

Parabola

A symmetrical U-shaped curve on graph representing a quadratic function

Zeroes

Also called roots are the x-intercepts of the parabola

Can have one, two or zero roots

Axis of Symmetry

A vertical line that divides the parabola into two equal halves

Axis of Symmetry is the sum of the roots divided by two

Y-intercept

The co-ordinate where the parabola crosses the y-axis

Optimal Value

The lowest or highest value/peak or bottom of the parabola depending on the a value - UPWARD or DOWNWARD?

Finite Differences in Quadratic Relations

Table of Values

FD = Subtracting consecutive y-values

If First Differences are constant, there is a linear relation

SD = Subtracting FD values

Solving Quadratic Equations

Factoring

Used to turn a quadratic into factored form to solve equation by finding X-intercepts

Finding what to multiply together to get an expression

Trinomial Standard Form
ax^2+bx+c
----------------------------
2x^2+10x-12
=2(x^2+5x-6)
=2(x^2-1x+6x-6)
=2(x(x-1)+ 6(x-1)
=2(x-1)(x+6)

1. Common Factor (If Any)
Common Factor = 2

Monomial Common Factor

12 x^2y-9x^3y^2z+18x^2y^2
_________________________
3x^2y
=3x^2y(4-3xyz+6y)

Identify greatest common factor = 3x^2y

Move GCF outside of bracket

Divide each term by the GCF and write what remains in bracket

Binomial Common Factor

2(x+1)-3y(x+1)
_____________
(x+1)
= (x+1)(2-3y)

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

m*n=-6
m+n=-5
-1*6=-6
-1+6=5

These two integers end up being (x-r) and (x-s) in factored form

Sub. in these numbers for b value

4. Factor by grouping

If the trinominal cannot be common factored and the a value is more than 1, you have to multiply a and c

m(n)=a(c)
m+n=b

x-1=0 x+6=0
x=1 x=-6

Perfect Square Trinomials

Same method of factoring is used as factoring trinomials except you write the product as the square of a binomial

a^2±2ab+b^2 = (a±b)^2

4x^2+12x+9
4(9) = 36
m(n)=36 m = 6
n+n=12 n = 6
Sub. in numbers as bx term = 12x
=4x^2+6x+6x+9
=2x(2x+3)+3(2x+3)
=(2x+3)(2x+3)
=(2x+3)^2

1. Factor out the GCF by dividing it with all terms

2. Move GCF outside of bracket

3. Multiply a value and c value
-Find two numbers that multiply to the product of a(c) and have the sum of b

4. Substitute the two numbers for the middle term

5. Group the terms with common factors and factor each binomial group

6. The last two binomials should be the same and so you simplify them together by writing the product as the square of the binomial

E.g. (2x – 3)(2x – 3) = (2x – 3)^2

Using this as a reference equation find out what a and b are

a^2 = 4x^2 so a=√4x^2 = 2x
b^2 = 9 so b=√9 = 3

Check that "2ab" is the middle term

2ab = 2(2x)(3)
= 4x(3)
= 12x

Therefore, 2ab is the middle term and that means it is a square trinomial

First and last terms are perfect squares

Difference of Squares

Identifying Characteristics

Contains Binomials (2-terms)

Contains a difference (Subtraction)

Contains Perfect Squares

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

Ax^2+Bx+C
=144p^2+0x-81
=144p^2-81
=(12p+9)(12p-9)

Using this as a reference equation find out what a and b are

a^2 = 144p^2 so a = √144p2 = 12p
b^2 = 81 so b = √81 = 9

Sub. in a and b into reference equation to get factored

Completing the Square

Convert Standard Form to Vertex Form and find Maximum/Minimum values or vertex of parabolas

y=3x^2-12x-5
y=(3x^2-12x)-5
y=3(x^2-4x)-5
-4/2=-2^2=4
y=3(x^2-4x+4)-5
y=3(x^2-4x+4)-17
y=3(x-2)^2-17

Vertex = (2,17) Minimum = -17

1. Put brackets around ax^2+bx terms

2. Common Factor the "a" value

3. Make Perfect Square Trinomial inside bracket using: (b/2)^2

4. Add the opposite sign of the (b/2)^2 inside the bracket
EX. 2(x^2+6x+9-9)+11

5. Move the opposite sign (b/2)^2 value outside bracket by multiplying it by "a" value

6. Add this value with the k value outside bracket to get final k value

7. Keep the "a" value outside bracket and square root the first term x^2, keep the sign of the middle term (addition or subtraction) and square root the last term in bracket, add a square outside of end bracket to give you: (x-h)^2

8. Write out the final Equation that is left over

Quadratic Equation

Finding x-intercepts from Standard Form using Quadratic Formula

X-intercepts/Roots/Zeroes = Solutions

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

2=-3x^2+4x+2

a=-3, b= 4, c= 2
2. Sub in a, b, and c values in formula

3. Solve for x1

4. Repeat except make the addition sign between -b and square root subtraction

5. Solve for x2

1. Make sure L.S or R.S is equal to 0

0=-3x^2+4x+2

(b^2-4ac)

Discriminant

Positive discriminant = 2 roots

If discriminant is zero there is 1 root

Negative discriminant = no real roots

Expanding Quadratic Functions

Vertex

The point where the axis of symmetry and the parabola meet at its maximum or minimum value

Optimal Value is the Y-Value of the vertex

Axis of symmetry on the x-axis is the x-value of the vertex

If Second Differences are constant, there is a quadratic relation

If FD or SD are not constant, the relation is niether

Standard Form

y = ax^2 + bx + c

a, b and c are all known values

a value cannot be 0

x is the unknown variable

c value is the y-intercept of parabola

Formula for Axis of Symmetry
--------------------
x = -b/2a
y=0.5x^2-2x-6
a=0.5 b= -2 c= -6

x= 2/2(0.5)
x= 2/1
x= 2

Factored Form

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

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

Solve each binomial individually for x to get the x-intercept(s)

y=0.5(x-6)(x+2)
--------------------------
x-6=0 x+2=0
x=6 x=-2
x1= (6,0) x2= (-2,0)

adding x-intercepts and then dividing by 2 = axis of symmetry - midpoint of x-intercepts

axis of symmetry is the x value of the vertex

(6,0) and (-2,0)
------------------
=[6+(-2)]/2
=[4]/2
=2
Axis of Symmetry
is (2,0)
X value of vertex
is 2 - (2,y)=vertex

Sub in axis of symmetry as x value to find optimal value (y value of vertex)

y=0.5(x-6)(x+2)
y=0.5(2-6)(2+2)
y=0.5(-4)(4)
y=0.5(-16)
y=(-8)

Optimal Value of parabola
is -8, meaning the y-value
of the vertex is -8

Sub in x as 0 to find y-intercept

y=0.5(x-6)(x+2)
y=0.5(0-6)(0+2)
y=0.5(-6)(2)
y=0.5(-12)
y=(-6)
Y-intercept is (0,-6)

Vertex Form

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

h = x value of the vertex and axis of symmetry

"h" represents a horizontal shift

"h" value is subtracted

k = y value of the vertex

"k" represents a vertical shift

"k" value is added

Sub in x=0 to find y-intercept

Base Parabola is
y=x^2

Vertex
---------------------------------------
(Axis of Symmetry, Optimal Value)
(x, y)
(2, -8)

Vertex
(h,k)

a = vertical stretch/compression factor

If a > 1 or a < -1, then the graph is stretched vertically by a
factor of a

If -1 < a < 0 or 0 < a < 1, (a is a fraction)then the graph is compressed vertically

Direction of Opening
-----------------------
If a>0 it is upward opening parabola, if a<0 it is downward opening parabola

Use Step Pattern for plotting points of parabola

Step Pattern
--------------
1,3,5,7,9,....

Multiply a value and step pattern to get correct points of a given parabola

Ex. a=2 Step Pattern
=2[1,3,5,7,9]
=2,6,10, 14, 18

2. Factor by Grouping (if necessary)

Group terms with common factors to solve

2x^2+6y+4x+3xy
=2x^2+4x+6y+3xy
=2x(x+2)+3y(2+x)
=(x+2)(2x+3y)

a^2 - b^2 = (a+b)(a-b)
144p^2 - 81 = (12p+9)(12p-9)

(12p+9)(12p-9) is a factored difference of squares

Expanding Binomials

(a + b)(c + d)
= ac + ad + bc + bd

e.g. (x + 4)(x – 3)
= x^2 – 3x + 4x – 12
= x^2 + x – 12

Perfect Square Trinomials

( x + a )^2 = x^2 + 2ax + a^2
( x – a )^2 = x^2 – 2ax + a^2

When the Binomial is square, to expand you must multiply the binomial by itself

( x + 5 )^2
= ( x + 5 ) ( x + 5 )
= x^2 + 10x + 25

1. Get rid of the square sign and have the binomials multiply eachother
( x – 3 ) ( x – 3 )

2. Use FOIL Method

Difference of Squares

( x + a ) ( x – a ) = x^2 – a^2

( x + 5 ) ( x – 5 )
= x^2 – 5x + 5x – 25
= x^2 – 25
Middle term cancel out

Sub. in x and a values to expand

x^2-5^2
=x^2-25

Distributive Property

a(b + c)
= ab + ac
e.g.
2(x + 4)
= 2x + 8

FOIL Method

First Outside Inside Last
( x + 3 ) ( x + 2 )
= x^2 + 2x + 3x + 6
= x^2 + 5x + 6