By: Dannia Kamakh
Properties Of A Quadratic
- The quadratic relation is called a parabola
- The vertex of the parabola is either the greatest point on the parabola or the smallest point
- If the parabola opens up it has a minimum
- If the parable opens down it has a maximum
- If the information is in a table then the second differences determines wether it will open up or down if the second difference is negative then it opens down if its positive it opens up
- The vertex's co-ordinates are (h,k) x=h and y=k
- The equation of the axis of symmetry is x=h
You first multiply the x in the first bracket with the x in the 2nd bracket and the number in the 2nd bracket then you repeat the process only using the 2nd number in the first bracket this time.
=x^2+3x+3x+9 *** The first number should always have a ^2 and there should also be 2 other x's***
Then you simplify and get:
a(b+c)= FactoringFor example: 10a+5a+40
You find a common factor between the first 2 numbers or 3 and divide every term by it.
Factoring By Grouping
so you first common factor it
Then you find what else they have in common (x+2) and factor it. And put whatever is in-between in brackets in the 2nd pair of brackets .
Factoring Simple Trinomials
So 6 x 1=6 (Now I have the product) now I need to figure out what 2 numbers add to +7 and multiply to 6?
6 and 1 do (6 x 1=6 and 6 + 1=7)
so i simply put them into the brackets with the x's
If I wanted to check and expand and simplify I should get the original equation back.
Factoring Complex Trinomials
=8x^2+10x+12x+15 (sub in the 2 numbers that multiply to get the P and add to get the S)
=2x(4x+5)+3(4x+5) (Factor by Grouping)
=(4x+5)(2x+3) (Factored Form)
Completing the Square
y=(x^2+6x)+11 (Block of first 2 terms)
y=(x^2+6x+9-9)+11 (Divide middle number by 2 and square it, make it equal to "0")
y=(x^2+6x+9)+11-9 (Factor inside the bracket)
y=(x+3)^2+2 (Vertex Form)
The Quadratic Formula
Graphing and Solving Quadratic Equations
1. You need to factor it by completing the square
2. You now have your factored form quadratic equation no we can find the x and y intercepts
y=(x+1)(x+3) (To get your x-int you need to make each bracket =0)
to get you y-int you need to make your x's equal to 0
3. To find your vertex you need to get your x's add them together and divide them by 2 this will give you your h
Now you take your h and plug it back into the original equation where the x's are
Answer: x=-1,-3 y-int=3 vertex:(-2,-1)