Dealing with Quadratics!
Alerbra Guide sheet
whats a quadratic?
-has the highest degree of 2
-graphed as a parabola
Different Quadratic Forms
Standard: (y=ax^2 +bx+c)
-reveals little about the parabola
-a =determines whether the parabola opens up or down also whether its a vertical stretch or compression.
-c =he y intercept (where the parabola touches the y-intercept)
-to change into factored you factor and to change into vertex you complete the square.
Factored: y=a (x-r)(x-s)
-reveals little about the parabola.
-a =determines whether the parabola will open up or down or whether it's a vertical stretch or compression.
- r,s= reveals the zeros ,x-intercepts (when the parabola will touch the x-axis)
-to change into standard you expand and simplify and to change into vertex, from there you complete the square.
Vertex: y=(x-h)2 +k
-reveals the most about the parabola
-a= determines weather the parabola opens up or down and whether its a vertical stretch or compression.
-h= x-value of the maximum or minimum (when does it happen?)- the time it takes (left or right).
-k= the maximum or minimum (where does it happen?)
How to Expand and simplify :
EXAMPLE: the equations seen below is in factored form expand and simplify.equation:y=a(x-r)a(x-s).
step1: First to expand you, apply everything directly in front of the bracket into every term inside the bracket.-y=a(x-r)a(x-s)-y=ax-ar+ax-s.
step2: You collect the like terms to simplify ,meaning collect everything that has the same components. (ex; coefficients,exponents ect)-y=ax-ar+ax-sy=2ax-ar-s.
step3: Leave what ever cannot be collected and simplified and just add them into the equation.
How to Compleate a Square:
(Ax2+Bx+C)- Referance Equation.step1:Block off the first two terms. 1)y= 2x2+8x-3 then... y= (2x2+8x)-3.
step2: Factor out the A value. 2)y=2(x^2+4x)-3. (note : when you factor out the A you divde the middle term by it.)
step3:To find your zeros didvide te middle term by two the square it .y=2(x2+4x+4-4)-3.
step4:Take the negative out then solve the ending and divide the equation by two. 4)y=2(x+2x)2-11.(note:when you take the negative out multiply it by the first term "A" then take out the positive zero and bring the square root to the end bracket.)
Quadratic Formula
Factoring
There are many ways of factoring here are just SOME ways!
GOLDEN RULE:
-always look for a common factor
-then check if it can be factored any more
-Difference of squares.
here is an example: x^2-9 (you would square both terms)
another example is: (x+5)(x-5)
these are the two types!
-Decomposition.
here is an example: 2x^2+5x-7
-start off by solving for the product and the sum for the middle term and replace it: 2x^2+7x-2x-7
-then factor out common terms :x(2x+7)-1(2x+7)
- then simplify even more (2x+7)(x-1)
-By Grouping
here is an example: x^3+2x^2+8x+16
-the first thing to do would be to block off each two terms then factor out what ever is in each bracket x^2(x+2)(X^2+8)
-then factor/simplify the rest! (x+2)(x^2+8)