Quadratics
By:Puneet Guleria
Intro
Table of context
Quadratics in vertex form
Key features of quadratic relations
Scatter plotLine/curve of best fit
Finite differences
-linear relation
-quadratic relation
-neither
Transformations of quadratics
-writing equations given the transformation
Graphing using step pattern
Finding equations given the vertex
Quadratics in factored form
Common Factoring
Factoring Trinomials
Factoring Difference of Squares and Perfect Square Trinomials
Solving Quadratics by Factoring (finding the zeros)
Graphing Quadratics in Factored Form
Quadratics in standard form
The Quadratic Formula
Key features of quadratic relations
Axis of symmetry(where the parabola is divided equally)
Optimal value(the y coordinate of the vertex, can be maximum or minimum)
Zeros/Roots(the x-intercepts)
Scatter plot
Line and Curve of best fit
Step2: make the line/curve of best fit in a place where almost all points on both side of the line are equal. this is how to determine and make a line or curve of best fit.
line of best fit
Curve of best fit
Using finite differences to determine if the relation is linear, quadratic, or neither
Linear relation
Quadratic relation
Transformations
writing equations given the trasformation
The steps
1- identify weather your parabola is opening up or down.
2- identify the stretch or compression(a)
3- identify the transformation left or right (h)
4- identify the transformation up or down (k)
5- finally put your variables into the equation y=a(x+3)^2+3
Graphing using the step pattern
Finding an equation given the vertex
steps
1- sub in your vertex into your equation y=a(x-h)^2+k (plug x value as your h but make your x the opposite sign(negative-positive or positive-negative) then for k plug in your y value the same as it is)
2- sub in one of your x intercepts into the equation
3- solve until you find the value of (a) which will complete the equation
4- plug in a to you original equation to complete the equation
Finding x and why intercepts in vertex form
multiplying binomials and special products
(x+5)^2---(x+5)(x+5)----x^2+5x+5x+25---x^2+10x+25 formula=a^2+@ab+c
(x+5)(x-5)---x^2+5x-5x-25----x^2-2 formula where equations are same but middle sign is different a^2-b^2
common factoring
for example
6x^2 − 2x
whats common is 2 and x so the new term would be
2x(3x-1)
factoring trinomials
Step 1 find two numbers that multiply to give ac (in other words a times c), and add to give b.
Step 2 Rewrite the middle with the new numbers to form a equation with 4 terms
Step 3 Factor the first two and last two terms separately
Step 4 put the common factors together
here is an example
2x^2 + 7x + 3 sum of (a)(c)=6 (6)(1)=6 6+1=7
2x2 + 6x + x + 3
2x(x+3) + (x+3)
(2x+1)(x+3)
factoring differences of squares
1 look for a gcf
2 square root the two numbers
3 determine if the the new numbers can be factored further
4 put numbers into factored form (x+a)(x-a)
Example: 18x^2-98y^2
2(9x^2-49^2)
square root the middle terms
2(3x-7y)(3x+7y)
factoring perfect square trinomials
factoring overview
solving quadratics by factoring
so the first step is to get one side to be zero- x^2-10x=-16---x^2-10x+16=0
next find two numbers that multiply to give ac (in other words a times c), and add to give b-
(x-8)=0 (x-2)=0
then solve for zeros-- x-8=0--- x=8 x-2=0--- x=2