Math 101
Learn everything for math here
contents
chapter 4.2 quadratic functions
Chapter 4.3 Transformations of Quadratics
4.4 graphing transformation of quadratic relation
4.5 finding roots/zeroes/x intercepts from vertex form
5.1 multiply polynomials
5.2 special products
chapter 5.3 common factoring
chapter 5.4 simple trinomial factoring
chapter 5.5 complex trinomial factoring
chapter 5.6 factoring special products
6.1 completing the squares
chapter 6.2 and 6.3 solving quadratic equations in factored form
6.4 quadratic formula
chapter 4.1 Investigating Non-Linear Relations
- a parabola can open down or up
- the vertex is the point where the axis of symmetry connect with the parabola
-its like the middle of the parabola and the highest point or if its a negative parabola which has a opening down, it would be the lowest point
- it also divides the parabola into two halves
- there is a term called zeroes where the parabola crosses the x axis
- other names for zeroes are roots or x intercepts
chapter 4.2 quadratic functions
a,b,c are real numbers where a=0
e.g 4x2+10x+7
to make an parabola you need to make a chart it will be easier
y=x2
x y
-2 4
-1 1
0 0
1 1
2 4
e.g y=-4x2+10x=7 - the opening will be down.
How do you know if its an quadratic relation or not?
if the second differences of the chart are constant then it is a quadratic relation and if the first differences are constant then it is an linear relation.
Chapter 4.3 Transformations of Quadratics
y and x do not change
terms:
a- determines how narrow or wide the parabola is
h- the x coordinate of the vertex
k- the y coordinate of the vertex
also:
a= the vertical stretch or compression by a factor of a/ reflected in the x axis if a is negative
h= the horizontal transition
k= the vertical transition
this is an basic equation:
4.4 graphing transformation of quadratic relation
when a number is subtracted from x2 the graph goes downwards on the y axis.
when x2 is multiplied with a number between 0 and 1 the width of the parabola becomes wider.
when the coefficient of x2 is negative the parabola opens downwards. making a frown.
the difference of y=a(x-3)2+4
1. opening upwards
2. translated 2 units to the right
3. translated 4 units up
y=(X+5)2=2 the vertex is (-5,2)
4.5 finding roots/zeroes/x intercepts from vertex form
vertex form y=a(x-h)2+k
5.1 multiply polynomials
2 terms- binomial
3 terms- trinomial
to simplify a polynomial you need use collecting like terms and distributive property.
(x+5) (x+6) 1. distributive property
x2+6x+5x+30 2. collect like terms
x2+11x+30
5.2 special products
1. square the first term
2. 2 times the product of the term
3. square the last term
example:
(x+6)2 x+6 x+6
x2+2(x)(6)+62
x2+12x+36
product of sum and difference
when the sum and difference of two terms are multiplied, the two middle terms cancel each other out so they are opposite
example:
(x+7) (x-7)
x2+-7x+7x+49
(x)2-(7)2
factoring
factoring is different than expanding, factoring looking for a expression to multiply while expanding has to do with just multiplying.
example
chapter 5.3 common factoring
method #1: look for the greatest common factor
gcf aka greatest common factor
gcf= -5y gcf= -4y
factoring a monomial
1. find gcf for both coefficient and variable
2. divide all the terms by gcf
example
2x+8y 5x+15x
2(x+4) 5x(x+3)
factoring a binomial
if there are 2 binomials that are the same then that is a binomial common factor.
2x(x+4)3(x+4) =(2x+3) (x+4)
4x(2x+3)2(2x+3) =(4x+2)
factoring by grouping
group the common factors into 2 terms to produce a binomial factor.
example
ax+ay+4x+8y
(ax+ay)+(4x+4y)
a(x+y)+4(x+y)
(a+4)+(x+y)
chapter 5.4 simple trinomial factoring
a complex trinomial is when a b c are more than 1
if given a quadratic in standard form you factor it to make it factored form and the other way around
standard form
factored form
chapter 5.5 complex trinomial factoring
not all quadratics can be factored.
1. before anything look at the common factor first when factoring a trinomial.
2. find to numbers that multiplied together equal a and c but added together equal b
3. then check up the middle term and factor by grouping
example
6x2+5x+1
(6x2+2x)(3x+1)
2x(3x+1)1(3x+1)
chapter 5.6 factoring special products
(a+b)2 a2+2ab+b2
(a-b)2 a2-2ab+b2
example
(x-6)2
x2-2(x)(6)+62
x2-12x+36
how to get a quadratic in standard form to factored form.
1. find 2 numbers that add up b and multiplied together equal.
2. if b and c are positive then r and s will be positive but if. if c is poistive and b is negative and one is positive then both r and s are negative but if c is negative then either r or s will be negative.
example
x2+7x+12
(x+4)(x+3)
x2+8x-12
(x+6)(x-2)
6.1 completing the squares
1.put equation into vertex form
2. group x terms
3. divide coefficient of middle term by 2 then square it then add and and subtract number inside bracket
4. remove subtracted term from brackets
5. factor brackets as perfect square trinomial
example
y=x2+8x+4
y=(x2+8x+16)-16+4
y=(x2+8x+16)-12
y=(x+4)2-12
completing the squares when the value of ''a'' is more than 1
step
1.group the x terms.
2. factor the value of ''a'' from the x intercepts.
3. divide coefficient by of middle term by 2 then square it and then add and subtract that number inside the bracket.
4. remove negative term from brackets and multiply it by the a value that you factored.
5. write the perfect square trinomial as a perfect square binomial.
example
y=2x2+8x+13
y=2(x2+8x+16)-16+13
Y=(x2+8x+16)-32+13
Y=(x2+8x+16)-19
Y=(x+4)2-19
you try
y=x2-6x+4 ANSWER-Y=(x-3)2-5
chapter 6.2 and 6.3 solving quadratic equations in factored form
solving-finding the zeroes/x-intercepts/roots.
1. make one side zero and then make all the brackets to zero. = (x+r)=0 (x+s)=0
2. then solve for x
example
6x2-15=0
(6x2+9x)(-10x-15)
3x(2x+3)-5(2x+3)
(2x+3)(3x-5)=0
2x=-3 3x=5
x=-3/2 x=5/3
x=1.5 x=1.7
the two x intercepts are (1.5,0) (1.7,0).