Quadratics
By: Mohit Devassar
Introduction
Table of content
- Step Pattern
- First differences
- Second differences
- Vertex Form
- Optimal value
- X-intercepts/zeros
- Transformations
Factoring
- Trinomial Factoring
- Binomial factoring
- Algebra Tiles
- Factoring by grouping
- Factor form
- Perfect squares
- Square differences
- Standard form to factored form
Solving
- Standard form to vertex form
- Quadratic formula
- What is the quadratic formula
- Discriminant
- Solving from vertex form
- Solving from standard form
- Word problem
- Reflection
Bridges
Rainbow
Dolphin
Parabolas vertex form
Step Pattern
Graphing parabola in vertex form
First Differences
Second Differences
Vertex Form equation
Vertex Form
Axis of symmetry
Optimal Value
How to graph with Vertex form
X - intercepts/zeros
Example:
y = 2(x+2)^2-8
0 = 2(x+2)^2-8
8 = 2(x+2)^2
8/2 = (x+2)^2
4 = (x+2)^2
+-2 = x+2
-2 = x+2
-2-2 = x
-4 = x
Transformations
The A value controls if the parabola open up or down.
The H value controls the horizontal movement.
The K value controls the vertical movement.
All of these values must be used when using transformations.
If the A value is greater then 1 then the parabola is being stretched and if it is smaller then 1 then the parabola is being compressed and if it is 0 then nothing changes.
If the K value is increased then it would move up by that many numbers and if it is decreased then it would move down by that many numbers.
If the H value is increased then the it would move left by that many numbers but if it is decreased then it move right by that many numbers
FACTORING
Algebra Tiles
Factoring Binomials
= 2(x+2)
I got this by finding the greatest common factor (GCF) and putting it to the side and dividing everything by the GCF which got me what was left in the bracket. You can check if your answer is right by distributing the 2 back in the bracket. It should equal back to the original equation.
Check:
2(x+2)
= 2x+4
Factoring trinomials (complex) (simple)
2x+4x+16x
= 2x(2+8)
Simple:
6x+15
=3(2x+5)
The process is the exact same as factoring Binomials but there is just three numbers that is why it makes it trinomials. A simple equation does not have a a value. (x+4x+16x) but you would factor it the same way.
Factoring by grouping (simple) (complex)
2x(x+3)+5(x+3)
= (2x+5) (x+3)
Simple:
x(x+2)+4(x+2)
= (x+4) (x+2)
All you have to do is find something common like (x+3) and (x+3) and make them represented by one binomial instead of two and then just group your remaining numbers in this case it was (2x+5) and then make it into a factored form equation by putting them beside each other like in the example.
Factoring standard form
Factored Form y=a(x+a)(x+b)
Factoring Perfect squares
A perfect square is any trinomial that follows a^2+2ab+b. Another way the perfect square equation can look is ax^2+b^2. If this is the form, make sure all coefficients are square rooted. An example to this would be turning 9x^2+81 into 3x^2+9^2.
Differences of Squares a2-b2
Standard Form to Factored Form
Factored form to Standard form
Solving
Standard form to vertex form
Quadratic formula
What is the quadratic formula
Discriminant
The Discriminant is the number inside the square root of the equation.
1) If the Discriminant is negative then there is no solution.2) If the discriminant is 0 then there is 1 solution.
3) If the Discriminant is positive then there are 2 solutions.
Solving from vertex form
Solving from standard form
Word problem (example)
a) what are the dimensions of the sides?
b) what is the maximum area?