unit 2 factoring and quadratic...
TRANSCRIPT
Unit 2 Factoring and Quadratic Functions
Unit 2
Factoring and Quadratic Functions
Topic 1
Exponents
Unit 2 Factoring and Quadratic Functions
Practice:
Rewrite and/or evaluate using exponents:
1) 4 * 4 * 4 * 4
2) y * y * y * x * x * x * x * x
3) (5)2
4) (1/2)4
5) (4)3
A ____________ is a number, a variable or a product of a number and one or more variables.
ex. Non ex.
xn is called a power and represents the result of multiplying x by itself n times. x is called the ______, and n is the ________ or ________.
Unit 2 Factoring and Quadratic Functions
To multiply two powers with the same base, add their exponents.am * ap = am+p
1) (6n3)(2n7)
2) (3pt3)(2t4p3)
To find the power of a power, multiply the exponents.(am)p = am*p
3) (b3)5
4) [(23)2]4
To find the power of a product, find the power of each factor and multiply.(ab)m = ambm
5) (2xy3)5
6) (2y4z2)3
Unit 2 Factoring and Quadratic Functions
HW Page 1761240 even and 41
HW Ans
12. 3214. 15,62516. 1,02418. a820. x4y622. r3s324. 54 212 = 2,560,00026. a11
28. yc+230. xm+132. True34. False36. False38. False40. 32 persons, 6th meeting41. 105 or 100,000 cm
Unit 2 Factoring and Quadratic Functions
To divide two power with the same base, subtract the exponents.am = ampap
7) c11 c8
8) k7m10p k5m3p
To find the power of a quotient, find the power of the numerator and the power of the denominator. a m = am b bm
9)
10)
A zero exponent is any nonzero number raised to the zero power and is equivalent to ________.
For any nonzero number a and any integer n, an is the reciprocal of an. an = 1
an
11) 24
12) 1 f4
Unit 2 Factoring and Quadratic Functions
For any nonnegative real number b, b½ = √b
13) 25½
For any positive real number b and any integer n>1, b1/n = n√b
14) 81/3
15) 1296¼
For any positive real number b and any integers m and n>1, bm/n = (n√b)m or n√bm
16) 642/3
17) 363/2
Unit 2 Factoring and Quadratic Functions
For any real number b > 0 and b ≠ 1, bx = by if and only if x = y.
18) 5x = 53
19) 36x1 = 6
20) 25x = 82x4
Scientific Notation Review:
a x 10n where 1 < a < 10 and n is an integer.
If n is positive then the number is ___________________.
If n is negative then the number is ____________________.
1) 201,000,000
2) 0.000051
3) 6.32 x 109
4) 4 x 107
Unit 2 Factoring and Quadratic Functions
Evaluate: (Use your calculator)
a) (3.5 x 103)(7 x 105)
b) (3.066 x 108) ÷ (7.3 x 103)
c) (2.4 x 104) + (3.532 x 106)
d) (4.789 x 107) (2.345 x 105)
e) Order from least to greatest: 5.46 x 103, 6.54 x 103, 4.56 x 104, 5.64 x 104, 4.65x 103
Unit 2 Factoring and Quadratic Functions
Topic 2
Polynomials
A ____________ is a monomial or the sum of monomials, each called a term.
A ____________ is the sum of ____ monomials.
A _____________ is the sum of ___ monomials.
Monomial:
Binomial:
Trinomial:
Unit 2 Factoring and Quadratic Functions
Degree Name
0 constant
1 linear
2 quadratic
3 cubic
4 quartic
5 quintic
6 or more 6th degree, 7th degree, and so on
The standard form of a polynomial has the terms from in order from greatest to least degree.
ex.
To add or subtract polynomials, add terms with the same degree.
1) (2x2 + 5x 7) + (3 4x2 + 6x)
2) (y4 3y + 7) + (2y3 + 2y 2y4 11)
3) (3x2 + 2x 6) =
4) (3 2x + 2x2) (4x 5 + 3x2)
5) (4x3 3x2 + 6x 4) (2x3 + x2 2)
Unit 2 Factoring and Quadratic Functions
Multiplying Polynomials:
By a monomial:
1) 3x2(7x2 x + 4)
2) 6d3(3d4 2d3 d + 9)
Binomial times a binomial:
3) (2x + 3)(x + 5)
4) (x 2)(3x + 4)
Unit 2 Factoring and Quadratic Functions
Any Polynomial:
5) (6x + 5)(2x2 3x 5)
6) (3x 5)(2x2 + 7x 8)
7) (m2 + 2m 3)(4m2 7m + 5)
HW
Page 181 Numbers 42, 44, 46, 51
Page 185 Numbers 2533 odd
Unit 2 Factoring and Quadratic Functions
Topic 3
Factoring
Factoring is the process of expressing polynomials as the product of other polynomials. (Think of it as working backwards)
Factoring using GCFa) 27y2 + 18y
b) 4a2b 8ab2 + 2ab
Unit 2 Factoring and Quadratic Functions
Factoring by grouping:a) 4qr + 8r + 3q + 6
b) rn + 5n r 5
Undoing the double distribution:x2 + bx + c = (x + m)(x + p) when m + p = b and mp = c
a) x2 + 6x + 8
b) x2 + 7x + 12
Unit 2 Factoring and Quadratic Functions
HWpg448
1823pg 461310
HW on your desk.
Take out a sheet of loose leaf and place your name on it. (this is not a quiz)
Unit 2 Factoring and Quadratic Functions
Some general notes:
1) always try _GCF_ first.
2) If you have a polynomial set up your parentheses.
3) If c is _Positive_ then the signs in the parentheses are the same______.
4) If c is _Negative__ then the signs in the parentheses are Different.
Factoring ax2 + bx + c, where a ≠ 1.Factor by grouping.
a) 5x2 13x + 6
b) 2x2 + 5x + 3
Unit 2 Factoring and Quadratic Functions
Special case:a2 b2 = (a +b)(a b)
a) x2 25
b) x2 64
More practice:
a) 12x + 24y
b) x2 8x + 15
c) 3x2 6x 45
d) 64 25x2
e) 9y2 12y + 4
f) x4 1
HW on your desk
Unit 2 Factoring and Quadratic Functions
CW from Friday on your Desk!!
Unit 2 Factoring and Quadratic Functions
Topic 4
Solving Quadratic Functions
HW in the basket
Zero Product Property:If the product of two factors is 0, then at least one of the factors must be 0.If ab = 0, then a = 0, b = 0 or both a and b equal zero.
a) x2 3x = 0
b) (2d + 6)(3d 15) = 0
Unit 2 Factoring and Quadratic Functions
We can use factoring to solve polynomial equations.Be sure to get the equation equal to zero first!!
a) x2 + 10x + 9 = 0
b) x2 120 = 7x
a) 2x2 + 9x 18 = 0
b) 3x2 + 26x = 16
c) 4x2 + 19x = 30
Unit 2 Factoring and Quadratic Functions
a) 81 x2 =0 25
b) 9x2 81 = 0
c) x2 + 12x + 36 = 0
d) 4x2 24x + 36 = 0
HW page 507
numbers 12 20
Unit 2 Factoring and Quadratic Functions
Classwork day!!!
Page 508 Numbers 4548
HW pg 508 numbers 4950
Quiz Thursday/Friday
Topic 5
Graphing Quadratic Functions
Unit 2 Factoring and Quadratic Functions
A quadratic function is a nonlinear function that can be written in the form f(x) = ax2 + bx + c, where a ≠ 0.
These graphs are called parabolas.
Graph:f(x) = 2x2 + 4x 3
x f(x)
Unit 2 Factoring and Quadratic Functions
A parabola is symmetric about a central line called the axis of symmetry.
The formula for the axis of symmetry is
The axis of symmetry intersects a parabola at only one point, called the vertex. (The vertex could either be a minimum or a maximum.)
If it is a minimum then the a value is positive. The parabola opens up.
If it is a maximum then the a value is a negative. The parabola opens down.
The xintercepts of the graph are called the "roots" or "zeros" of the function.
Graph f(x) = 2x2 + 4x 3
x f(x)
Unit 2 Factoring and Quadratic Functions
Completing the square!!
Factored Form Write the Factors Distribute Standard Form
(x + 1)2 (x +1)(x+1)
(x + 2)2
(x + 3)2
(x + 4)2
(x + 5)2
(x + 20)2
Unit 2 Factoring and Quadratic Functions
Standard Form Factored Form
x2 + 12x + 36 (x + 6)2
x2 12x + 36
x2 + 20x + 100
x2 3x + 9 4
x2 + 100x + 2500
Try going from standard form to factored form.
What about x2 + 8x + 3?
Unit 2 Factoring and Quadratic Functions
Lets try some more. a) a2 4a + 15 b) n2 2n 15
c) x2 1,000x + 60,000 d) x2 bx + c
Vertex form is ___________________________
Solving an equation by completing the square:
1) x2 + 2x = 8 2) x2 2x = 12
3) 12 = x2 + 6x 4) x2 = 1 0.6x
Unit 2 Factoring and Quadratic Functions
Transformations with Parabolas.
Translation: sliding of the original functionGraph: f1(x) = x2
Graph: f2(x) = (x 3)2
Graph: f3(x) = (x + 4)2
Graph: f4(x) = x2 5
Graph: f5(x) = (x + 2)2 + 3
Transformations (continued):
Dilation:
Graph: f1(x) = x2
Graph: f2(x) = 5x2
Graph: f3(x) = 0.25x2
Unit 2 Factoring and Quadratic Functions
Transformations (continued):
Reflections:
Graph: f1(x) = x2
Graph: f2(x) = x2
Combinations:
Graph: f1(x) = x2
Graph: f2(x) = 3(x + 4)2 6
Quadratic Formula!!
When a function is not factorable the quadratic formula can be used to find the roots of the function.
The quadratic formula is x =
This is provided to you on your reference sheet!! (YAY!!)
Use the quadratic formula to solve the equation:
a) 4x2 3x + 5 = 0 b) 2x2 = 4x 3
Unit 2 Factoring and Quadratic Functions
The value under the radical is called the __________________________.
If the discriminant is a positive perfect square then there are _______________________________________________.
If the discriminant is negative then _______________________________.
If the discriminant is not a perfect square then there are ________________________________________________________.
If the discriminant is zero then there is _______________________________.