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Accelerated Math I
Unit 3 (6 weeks)-State Framework is Unit 5
Lesson 1-Polynomials
1) Computation with Polynomials:
a. Add & Subtract like terms
2 xy  3x
b. Divide w/monomial:
x
c. Multiply with a focus on binomials using the distributive property:
(x+3)(x-4)
2) Factoring:
a. Area & Volume Models for concrete examples of multiplying integrated
in the tasks
i. Algeblocks
ii. Algebra Tiles
b. Grouping
i. 3x+6+2x+4 = 5x+10 = 5(x+2)
ii. 3xy+9+2x²y+6x = (3xy+9)+(2x²y+6x) = 3(xy+3)+2x(xy+3) =
(3+2x)(xy+3)
c. Trial & Error
i. x²+4x+3 = (x+3)(x+1)
d. Special Products
i. (x+y)² = x²+2xy+y²
ii. (x-y)² = x²-2xy+y²
iii. (x+y)(x-y) = x²-y²
iv. (x+a)(x+b) = x²+(a+b)x+ab
v. (x+y)³ = x³+3x²y+3xy²+y³
vi. (x-y)³ = x³-3x²y+3xy²-y³
Lesson 2-Solving Quadratics
1) Review of transformations of f(x)=x²
a. ax²+c
i. x²-4
ii. x²+4
iii. 4x²
iv. ¼x²
b. Shifts
i. f(x)=a(x-h)²+k
ii. a=stretches/shrinks
iii. h=moves left/right
iv. k=moves up or down
2) 2 Forms of Trinomials:
a. Vertex Form
i. f(x)=a(x-h)²+k
b. Standard Form
i. f(x)=ax²+bx+c
c. Characteristics
i. Real Solutions: 1, 2, none
ii. Domain & Range: inequality intervals
b b
iii. Vertex: (h,k) or (
,f(
))
2a 2a
b
iv. Axis of Symmetry: x=h or x=
2a
v. Zeros: x-intercepts (x,0)
vi. Intercepts: x (x,0) & y (0,y) intercepts
vii. Extremes: maximum or minimum (vertex) unless domain is
restricted
1. Restricted domains
a. Area to work in/length of fence
2. a=open up; -a=open down
viii. Intervals or increase/decrease
ix. Rates of Change: slope between 2 points
d. Examples
i. (x+3)²: shifts graph left 3; Vertex (-3,0); Axis of Symmetry x=-3;
Zeros: x=-3; y-int=9;Minimum=(-3,0); 1 solution
ii. (x-3)²: shifts graph right 3; Vertex (3,0); Axis of Symmetry x=3;
Zeros: x=3; y-int= 9; Minimum=(3,0); 1 solution
iii. 4(x+3)²: shifts graph left 3 & stretch 4; Vertex (-3,0); Axis of
Symmetry x=-3; Zeros:x=-3; y-int.=36; Minimum=(-3,0); 1
solution
iv. -¼(x+3)²: shifts graph left 3, shrinks ¼ & flips down; Vertex= (3,0); Axis of Symmetry x=-3; Zeros: x=-3; y-int= -2.25;
Maximum=(-3,0); 1 solution
v. 4x²+24x+37 = 4x²+24x+36+1 = 4(x+3)²+1; shifts graph left 3,
stretches 4, and shifts up 1; Vertex=(-3,1); Axis of Symmetry x=3; Zeros=none; y-int=37; Minimum=(-3,1); no real solution
Lesson 3-Quadratics-More Graphing
1) Bigger “a” values
a. Practice of Lesson 2
2) Inequalities
a. x²+3x +2>0
i. (x+1)(x+2)>0
1. Roots: x=-1 or x=-2
2. x<-2 or x>-1
b. x²+6x+7≤-2
i. x²+6x+9≤0 = (x+3)²≤0
1. x≤-3
Lesson 4-Complex Quadratics
1) Quadratic Formula:
a.
b  b2  4ac
2a
i. x²+6x-2
6  62  4(1)(2)
1.
= 3  11 = the roots
2(1)
2) Discriminate:
a. = b 2  4ac
b. x intercepts/solutions
i.
 = imaginary number
1. roots=2 imaginary solutions
2. real roots = no real solution
0 =1 solution
ii.
iii.
 =2 real solutions
3) Imaginary Numbers:
a. imaginary number (i)
b.
1 =i, i²= 1 ( 1 )=-1
c.
16 = 4i ; 5 = i 5
d. Computation:
i. 6+7i+2i = 6+9i
ii. 6(7i) = 42i
iii. (3+2i)(4+7i) = 12 + 29i +14i² = 12 + 29i -14 = -2+29i
3  2i 1  i 3  3i  2i  2i 2 2(1)  i  3 5  i
iv. Conjugate:
(
)=


1 i 1  i
(1  i)2
1  1
2
e. Apply to Quadratic Complex Solutions
b  b2  4ac
i.
2a
1. x²+6x+12
2.
6  62  4(1)(12)
= 3  i 3 = the roots
2(1)
Lesson 5-Connections
1) Rational Equations
1 1 x
2 x2
 


a.
2 x 2x 2x
2x
1
1
x 1
2
x3




b.
2 x 1 2x  2 2x  2 2x  2
1
1
x 1
x2
2x  3
c.




x  2 x  1 ( x  2)( x  1) ( x  2)( x  1) ( x  2)( x  1)
2) Pythagorean Theorem
a. a 2  b 2  c 2
i. 3²+b²=12² ; b²=135 ; b= 3 15