Quadratics
All you need to know about parabolas!
By: Krishna Patel
Overview of the Unit
Quadratics 1
· Intro to Quadratics
· Different parts of a parabola
· Vertex Form
· Finding equation and analyzing vertex form
· Factored form
Quadratics 2
· Multiplying Binomials
· Common Factors and Simple Grouping
· Factoring Trinomials with leading 1
· Factoring Trinomials without leading 1
· Factor Special Trinomials
· Solving by factoring
Quadratics 3
· Completing the square
· Solving from vertex form
· Quadratic Formula + Discrimnant
· More Quadratic and Discriminant
Quadratics 1
Introducing the Parabola

- Parabolas can open up or down
- The zero is also known as the x-intercepts or "roots"
- The axis of symmetry divides the parabola into two equal halves
- The vertex is the point where the axis of symmetry and the parabola meet displaying its maximum or minimum value
- The optimal value is the value of the y co-ordinate
Vertex: Is always located half way between the zeroes, and it is where the parabola meets, there can either be a minimum of a parabola or a maximum.
Optimal Value: It is also known as the y coordinate and is the highest point of the parabola
Axis of Symmetry: The axis of symmetry goes through the x axis and is also known as the x value
Vertex Form
Graphing Vertex Form
a= vertical compression or stretch
k= changes the y coordinate (up or down)
h= changes the x coordinate (left or right)
Always remember to describe a graph using SRT
Example: y= 2(x-3)^2+5
S: vertical stretch by a factor of 2
R; vertical reflection
T: Horizontal Translation to the right by 3 units (-3)
Vertical Translation up by 5 units (5)
Finite Differences
If the first differences are constant it is linear, if the second differences are constant it is a quadratic relation.

1) Graphing using Step Pattern
Ex. y= 1/2(x-3)^2-5
Step 2: Plot the vertex of the equation (3, -5)
Step 2: 1,1 and 2,4 are the steps you use for step pattern
Step 2: Multiply the a value of the equation (1/2) by the y values of (1-1) and (2-4)
Step 3: Move accordingly

2) Graphing using Mapping Notation
- Write a mapping formula
- Complete a table of values for y= x^2
- Determine the transformed "key" points
- Sketch the base y= x^2
- Sketch the new graph
- State possible x and y values

Graph the following using mapping notation and step pattern:
1) y= -0.5 (x+4)^2 -3
2) h= -4 (t-6)^2 + 19
3) A parabola has an equation of y= 6 (x+3)^2 + 4
a) Graph this equation
b) State the direction of opening
c) Write the x intercepts
Finding Equations and Analyzing Vertex Form
Step 1: y= a (x-h)^2 + k
Step 2 : y= a (x-2)^2 + 6
Step 3: 3= a (5-2)^2 + 6
Step 4: 3= a (3)^2 + 6
Step 5: 3= a (9) + 6
Step 6: 3= 9a+ 6
Step 7: 3-6 = 9a
Step 8: -3= 9a
Step 9: -3/9 = a
Step 9 (simplify the fraction): -1/3 = a
Therefore the equation is: -1/3 (x-2)^2 + 6
For more practice there are similar questions in the math workbook (chapter 4.4 page 42)

Factored Form
Find x-intercepts by setting y to 0 and the vertex is between the two x intercepts. For factored form the a is the stretch or compression, its a stretch if its a whole number and a compression if its a fraction.
Factored Form: y= a (x-r) (x-s)
Quadratics 2
Multiplying Binomials
- F: First
- O: Outer
- I: Inner
- L: Last
Example: (x+1) (x+3)
- F: (x) (x) = x^2
- O: (x) (2)= 2x
- I: (1) (x) = x
- L: (1) (3) = 3
x^2 + 2x + x + 3
x^2 + 3x + 3
Example 1:
(r-8) (r+3)
(multiply the first r by the second bracket, and the -8 with the second bracket)
r^2 + 3r - 8r -24
r^2 - 5r - 24
Example 2:
5(x+4) (x+6)
(distribute 5 to the first bracket only)
(5x+ 20) (x+6)
5x^2 + 30x+ 20x + 120
5x^2+ 50x+ 120
Squaring Binomials:
(a+b)^2 =a^2+ 2ab+ b^2
Example:
(x+4)^2
= (x)^2 + 3(x) (4)+ (4)^2
= x^2 + 8x + 16 (the answer is called perfect square trinomial)
(a+b)(a-b)= a^2 - b^2
Example:
(x+5) (x-5)
= (x)^2 - (5)^2
= x^2- 25 (the answer is called differences of squares)

Common Factoring and Simple Grouping
2x^2------> 2. x . x
4x------> 2. 2. x
GCF: 2x
2x 2x^2+ 4x divided by 2
2x (x+2)
Factoring Simple and Complex Trinomials
Product and Sum:
Example 1:
Factor x^2=7x+12
(Find a number that multiples to 12 and adds to give 7)
P= 12 (4 times 3)
S= 7 (4 plus 3)
Therefore: (x+3) (x+7)
Example 1:
Factor: x^2+7x+12
x^2+7x=12
x^2+ 4x+ 3x+ 12
x (x+4) + 3 (x+3)
(x+4) (x+3)
Complex Trinomials
Example: 6x^2 + 11x + 3
The first 2 numbers should multiply to 6, and the other 2 should multipy to give 3
When you multiply across and add the numbers they should add to 11.
3x---------1 ------> (3 times 3 = 9)
2x ---------3 ------> (2 times 1 = 2)
9+2 = 11
(3x+ 1) (2x+ 3)
6x^2+ 11x+ 3
Sum= 11 Product= 6 x 3= 18 (9, 2)
6x^2+ 9x+2x+3
3 (2x+3) + 1 (2x+3)
(2x+3) (3x+1)
Factor Special Trinomials
- a^2- b^2= (a+b) (a-b)
- Both a and be should be perfect squares
Example:
81x^2 - 64y^2
(9x)^2 (8y)^2
= (9x+8y) (9x-8y)
Example 2:
144y^2 - 169
(12y)^2 (13)^2
= (12y+13) (12y-13)
- The first and last terms are perfect squares, and the middle term is twice the product of the square roots
- a^2- 2ab+b^2 = (a+b)^2
Example 1:
16m^2 + 24m+ 9
(4m)^2+ 2(4m)(3)+ (3)^2
(4m+3)^2
This website helped me understand Perfect Square, I liked how they explained everything they did step by step and also gave reasons to why they're doing something in a specific way.
Solving by Factoring
(x+ 3) (x+4)= 0
x+ 3= 0 (x= -3)
or
x-4= 0 (x= -4)
therefore the x intercepts are x= -3 and x= -4
x^2+ 5x+6 = 0
P= 6 (3 times 2)
S= 5 (3 plus 2)
(x+3) (x+2) = 0
x+3= 0 (x= 3)
or
x+2= 0 (x= 2)
Therefore the x intercepts are x= 3 and x= 2
2x^2+ 14x+ 12= 0
(factor out the 2)
2 (x^2+ 7x+ 6)
P= 6 (6 times 1)
S= 7 (6 plus 1)
(x+6) (x+1)= 0
x+6= 0 (x= -6)
or
x+1= 0 (x= -1)
Therefore the x intercepts are x= -6 and x= -1
Quadratics 3
Completing the Square
Simple Practice
1) x^2 + 2x + _____ (Answer= 2/2^2 = 1)
2) x^2 + 8x +1____ (Answer= 8/2^2= 16)

Quadratic Formula and Discriminant
If the answer is greater than 0 there will be two solutions (roots)
If the answer is less than 0 there will be no solution (roots)
If the answer is 0 there will be 1 solution (root)

Quadratic Formula


Optimization


Sorry if the video is a little blurry