Quadratics Concept Map

Quadratics Concept Map

1. a curve used to represent a quadratic equation

   • definition
     • Parabola
       • vertex
         • the point where the axis of symmetry and the parabola meet, also the point where the parabola is at its maximum or minimum
       • axis of symmetry
         • divides the parabola into 2 equal halves
           • the 'x' value of the vertex
         • the sum of the two roots divided by 2
       • y- intercept
         • where the parabola crosses the y-axis
       • optimal value (maximum or minimum)
         • 'y' value of the vertex
       • direction of opening
         • determines whether the parabola faces upward or downwards
           • determined when 'a' value is either a negative or positive number
       • x- intercepts
         • points of the parabola that touch the x- axis
           • also called zeroes or roots
             • can be 1, 2 or no x-intercepts

2. Forms of Quadratic Equations

   • Factored Form
     • equation
       • y= (x-r)(x-s)
         • r &s re[resent x-intercepts
         • a represents direction of opening & vertical stretch or compression factor
         • example- y=2(x+5)(x-3)
           • opening: up stretch: a factor of 2
           • x-intercepts: x= -5 & x=1
   • Standard Form
     • equation
       • y= ax²+bx+c
         • formula for Axis of Symmetry
           • x= -b/2a
         • "c" value is the y-intercept
   • Vertex Form
     • equation
       • y= a(x-h)²+k
         • Vertex (h,k)
           • h= x value of the vertex and axis of symmetry
             • h value is always the oppoiste
             • "h" represents a horizontal shift
           • k= y value of the vertex
             • "k" represents a vertical shift
         • example- y=2 (x+3)²-8
           • opening: up stretch: factor of 2
           • vertex: (-3,8)
   • Transformations
     • Vertex Form
       • → If 'k' is positive a number, the graph shifts upwards → If 'k' is a negative number, the graph shifts downwards → If 'h' is a positive number, the graph shifts left → If 'h' is a negative number, the graph shifts right
     • Standard Form
       • → If a>0 it is a vertical stretch → If 0<a>1 it is a vertical compression

3. Finite Differences

   • table of values
     • First Differences
       • If First Differences are constant, there is a linear relation
     • Second Differences
       • If Second Differences are constant, there is a queadratic relation

4. Expanding

   • FOIL method
     • First Outside Inside Last
       • (x+3)(x+2) =x²+2x+3x+6 =x²+5x+6
     • Expanding Binomials
       • (a+b)(c+d) =ac+ad+bc+bd ex.(x+4)(x-3) =x²-3x+4x-12 =x²+x-12
     • Distributive Property
       • a(b+c) =ab+ac ex.2(x+4) =2x+8
     • Perfect Square Trinomials
       • ( x+ a)²= (x+a)(x+a) (x-a)²= (x-a)(x-a) When the Binomial is square, to expand you must multiply the binomial by iself
         • (x+5)² =(x+5) (x+5) =x² +10x+25
     • Difference of Squares
       • (x+a)(x-a) =x²-a²
         • (x+5)(x-5)

5. Solving Quadratic Equations

   • Method 1: Factoring
     • Simple Factoring
       • finding two numbers that add up to your 'b' value & multiply to give your 'c' value
         • x²+5x+6 = (x+3)(x+2) 3+2=5 3x2= 6
     • Common Factoring
       • Factor out GCF
         • 6m+15 → Find numbers that multiply into both 6 and 15 →1 and 3 →Find the greatest common factor; which is 3 →Write factored form so that when you expand you get the original expression =3(2m+5) →3x2m= 6m and 3x5= 15
     • Complex Factoring
       • finding 2 numbers that add up to the 'b' value and multiple to (a)(c) →2x²+9x+4 (4x2=8) →Find two numbers that add up to 9, and when multiplied give 8 →1x8=8 1+8=9 →rewrite expression, but replace 9x with the two numbers →2x²+8x+1x+4 →factor by grouping
         • Factor by Grouping
           • 2x²+8x+1x+4 →put brackets around two terms and factor =(2x²+8x)(+1x+4) =2x(x+4)+1(x+4)
   • Method 2: Quadratic Formula
     • Use this method when the equation is not factorbale
     • discriminant
       • x=b²-4ac
         • →when d=0, there will be 1 solution →when d>0, there will be 2 solutions →when d<0, there are 0 solutions
   • Method 3: Completing the Square
     • used when converting from standard form to vertex form
       • y=2x²+8x+6 y=(2x²+8x)+6 y=2(x²+4x)+6 y=2[x²+(4/2)²x]+6 y=2(x²+4x+4-4)6 y=2(x²+4x+4)-8+6 y=2(√ x²+4x+√ 4)-8+6 y=2(x+2)²-2

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