18071792
Description
Mind Map by isha sharma, updated more than 1 year ago
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Created by isha sharma
about 5 years ago
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Quadratics Concept Map
- 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
- 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 'x' value of the vertex
- the sum of the two roots divided by 2
- divides the parabola
into 2 equal halves
- y- intercept
- where the parabola crosses the y-axis
- where the parabola crosses the y-axis
- optimal value
(maximum or
minimum)
- 'y' value of the vertex
- '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
- determined when 'a' value is either a
negative or positive number
- determines whether the parabola faces
upward or downwards
- x-
intercepts
- points of the parabola that touch the x- axis
- also called zeroes or roots
- can be 1, 2 or no x-intercepts
- can be 1, 2 or no x-intercepts
- also called zeroes or roots
- points of the parabola that touch the x- axis
- vertex
- Parabola
- definition
- 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
- opening: up
stretch: a factor of
2
- r &s re[resent x-intercepts
- y= (x-r)(x-s)
- equation
- Standard
Form
- equation
- y= ax²+bx+c
- formula for Axis of Symmetry
- x= -b/2a
- x= -b/2a
- "c" value is the y-intercept
- formula for Axis of Symmetry
- y= ax²+bx+c
- equation
- 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
- h value is always the oppoiste
- k= y value of the vertex
- "k" represents a vertical shift
- "k" represents a vertical shift
- h= x value of the vertex
and axis of symmetry
- example-
y=2
(x+3)²-8
- opening: up
stretch: factor of
2
- vertex:
(-3,8)
- opening: up
stretch: factor of
2
- Vertex
(h,k)
- y= a(x-h)²+k
- equation
- 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
- → 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
- → If a>0 it is a vertical stretch → If
0<a>1 it is a vertical compression
- Vertex Form
- Factored Form
- Finite
Differences
- table of
values
- First Differences
- If First Differences are constant,
there is a linear relation
- If First Differences are constant,
there is a linear relation
- Second
Differences
- If Second Differences are constant,
there is a queadratic relation
- If Second Differences are constant,
there is a queadratic relation
- First Differences
- table of
values
- Expanding
- FOIL
method
- First Outside Inside
Last
- (x+3)(x+2)
=x²+2x+3x+6
=x²+5x+6
- (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
- (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
- 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
- (x+5)² =(x+5)
(x+5) =x²
+10x+25
- ( 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
- Difference of Squares
- (x+a)(x-a)
=x²-a²
- (x+5)(x-5)
- (x+5)(x-5)
- (x+a)(x-a)
=x²-a²
- First Outside Inside
Last
- FOIL
method
- 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
- x²+5x+6
= (x+3)(x+2)
3+2=5
3x2= 6
- finding two numbers that add up to your 'b' value
& multiply to give your 'c' value
- 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
- 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
- Factor out GCF
- 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)
- 2x²+8x+1x+4
→put brackets around two terms and factor
=(2x²+8x)(+1x+4)
=2x(x+4)+1(x+4)
- Factor by Grouping
- 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
- Simple Factoring
- 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
- →when d=0, there will be 1 solution
→when d>0, there will be 2 solutions
→when d<0, there are 0 solutions
- x=b²-4ac
- Use this method when the equation is not factorbale
- 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
- 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
- used when converting from standard
form to vertex form
- Method 1: Factoring
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