QUADRATICS

QUADRATICS

1. Formulas

   • Standard Form
     • y=ax^2+bx+c
       • Find y- intercept
         • "c"
       • Find zeros
         • Quadratic formula
   • Vertex Form
     • y=a(x-h)+k
       • Find vertex, and AOS
         • Optimal value
           • k
       • Translations/ Transformations
         • "a"
           • Stretches
             • lal>1
               • Vertically stretches by factor of lal
           • compression
             • 0<lal<1
               • Vertically compresses by factor of lal
           • Reflection
             • a<0
               • Reflected in x-axis
         • "h"
           • h>0
             • Horizontally translated to the right by lhl
           • h<0
             • Horizontally translated to the left by lhl
         • "k"
           • k>0
             • Vertically translated up by lkl
           • k<0
             • Vertically translated down by lkl
   • Factored Form
     • y=a(x-s)(x-t)
       • Find zeros/roots/x-intercepts
         • set y 0
         • Can find the vertex by Vx=(s+t)/2 and subbing the Vx in to the formula to find the Vy

2. Determining the number of roots

   • Using "a" and "K"
     • "a" and "k" have the same signs
       • No roots
     • "a" and "k" have different signs
       • two roots
     • "a" is irrelevant and k is not zero
       • One root
   • Quadratic formula
     • Define D=b^2 - 4ac as discriminant 1. D>0 (2 roots) 2. D=0 (1 root) 3. D<0 (No root)

3. Graphing

   • 1. Start with y=x^2 and transform all points
     • base formula
   • 2. Plot the vertex and use new step pattern for other points
     • New step pattern= original step pattern (1,3,5) times "a"

4. Completing squares

   • To convert an equation to a vertex form
   • y=ax^2 + bx +c
     • perfect squares
       • If the coefficient of x^2 is not 1
         • Take out the a before completing the square
       • If the coefficient of x^2 is 1 and c is square of half of b
         • y= (x + b/2)^2
   • word problems
     • Ask for max/min, or the x value that give the max/min

5. Factoring

   • 2 terms
     • difference of squares
       • ex) (x-1) (x+1)
     • not possible
       • When the constant is positive and the coefficient of the x^2 is 1
     • factoring
       • When the coefficient of x^2 is the constant or a factor of the constant
   • 3 terms
     • perfect square
       • (x+y)^2
     • complex trinomials
       • make sure factors are common first
       • ax^2 + bx + c
     • simple trinomials
       • no coefficients
         • no 'a'
           • x^2 +bx +c
     • use looping
   • 4 terms
     • grouping
       • First make sure to factor out the GCF
         • Group terms and factor

6. characteristics

   • Parabolic
     • Symmetrical
       • There is an Axis Of Symmetry (AOS) which is x= Vx
     • Curved line
   • table of value
     • The first difference is different
       • If the first difference is the same, the graph is linear
     • Second difference is the same
       • It means that the rate of change (the slope) is constantly changing
       • Half of the second difference is the "a" value of the equation
   • One vertex
     • represent maximum
       • "a" is negative
         • Concave down/ Opening down
     • represent minumum
       • "a" is postive
         • Concave up/ Opening up

7. Word Problems

   • Area questions
     • substitution
       • find one variable and sub into original equation
   • Number questions
     • Set up equations with the information given on the question
       • ex) The sum of two numbers is 60
         • Set x be one number and y be the other
           • x + y=60
       • Rearrange a equation to isolate a variable
         • Sub the rearranged equation to the other equation
           • Solve for a variable by completing square or using other methods
             • Sub the solved variable to an equation to find the value of the unsolved variable
   • Revenue questions
     • Find the max Revenue
       • Set x the number the # of increase/ decrease
         • Find the equation for cost and quantity and sub the equations to Revenue= cost * Quantity
           • Rearrange to vertex form and the k represents the max Revenue
     • Find the max profit
       • Profit= Revenue - cost*quantity
   • Projectile questions
     • optimal value
       • 'k'
         • Max height of projectile
       • Use completing square method to rearrange the equation to vertex form
     • When does it hit the ground?
       • Find the x- intercept
         • Factoring or quadratics formula
           • If one value is negative, it is inadmissible (If x represents time)
     • The initial value
       • "c"
         • Standard form

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