unit 2 factoring and quadratic...

Unit 2 Factoring and Quadratic Functions

Unit 2

Factoring and Quadratic Functions

Topic 1

Exponents


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 ________.


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


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


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


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


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


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


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:


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)


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)


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


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


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


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)


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


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


CW from Friday on your Desk!!


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


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


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


Classwork day!!!

Page 508 Numbers 4548

HW pg 508 numbers 4950

Quiz Thursday/Friday

Topic 5

Graphing 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)


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)


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


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?


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


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


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


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 _______________________________.

unit 2 factoring and quadratic...

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