Download Algebra 2 - CP Algebra 2 Overview

Semester A Notes
 The Real Number System
 Objective: Review the real number
system.
 Tools and Rules
 Real Numbers {R}: Any number that can be shown on the number line.
 Rational Numbers{Q}: Any number that can be written in the form a/b (where b ≠
0, and a and b are integers).
 Irrational Numbers{Q’}: Any number that cannot be written in the form a/b
where a and b are integers.
 Integer {Z}: A “whole” number (not a fraction or decimal) including positives,
negatives, and 0.
 Whole numbers: Same as set of integers, but not including Negative Numbers
 Natural Numbers{N}: Same as integers, but not including Negatives or 0.
 Natural Numbers:
 {1, 2, 3, …..}
 Whole Numbers
 {0, 1, 2, 3, ….}
 Integers:
 {…-2, -1, 0, 1, 2...}
 Rational Numbers
 Ex: 5, 0, -5/3, 7/9, 0.5, 4/1, 1.3333….
 (Includes repeating decimals and terminating decimals)
 Irrational Numbers:
 Ex: 𝟐, π, -2.5613402817…..
 (Includes non-terminating decimals)
 Why do we classify numbers?
 Have a 1-2 minute discussion with the person next to you, and write down
why you think having a system of classification is useful for math.
 Be prepared to discuss with the class.
 Objectives: To review the methods
used to simplify numerical
expressions and to evaluate
algebraic expressions.
 Tools and Rules
 PEMDAS
 |a| =
 a if a > 0 (+ #)
 0 if a = 0
 a if a < 0 (- #)
 Vocabulary
 Sum
 Difference
 Product
 Quotient
 Power
 Base
 Exponent
 Variable
 Expression
 (A) Pg. 11, #24
 (B) Pg. 11 #40
 Objectives: To review properties of
equality of real numbers and properties for
adding and multiplying real numbers.
 Tools and Rules/Vocabulary
 See properties of equality on pg. 14
 See properties of real numbers on pg. 15
Commutative Property (x and +)
Associative Property(x and +)
Identity Properties(x and +)
Inverse Properties (x and +)
 Opposite
 Reciprocal
 Distributive Property
 Combines x and + or –
 Multiplicative Property of 0
 Multiplicative Property of -1




 Pg. 18 #17
 (5) + (-9) =
 (5) – (9) =
 (-5) +(9) =
 (-5) + (-9)=
 (-5) – (-9) =
(+)(+) =
 (-5) – (9)=
(+)(-)=
(-)(-)=
(-)(+)=
(+)/(+)=
(+)/(-) =
(-)/(-) =
(-)/(+) =
 To subtract any number, add the
opposite
(-)(-)(-) =?
(-)^15 =?
(-) ^20 =
What’s the rule?
 Why is it important to have an order of operations when simplifying or solving
problems?
 Why do we need properties in math? How are they useful? (Extra: Can you think of
an analogy?)
 Have a 1-2 minute discussion with the person next to you, and write down your
conclusions.
 Be prepared to discuss with the class.
 Objectives: To review
properties of equality and
properties for adding and
multiplying real numbers.
 Vocabulary:
 Field- A set of numbers that has all
the properties listed

(The set of Real Numbers is a field.)
 Tools and Rules/Vocabulary
 See properties of equality on pg. 14
 See properties of real numbers on
pg. 15
Commutative Prop (x and +)
Associative Prop (x and +)
Identity Prop (x and +)
Inverse Prop (x and +)
 Opposite
 Reciprocal
 Distributive Prop




 Combines x and + or –
 Multiplicative Prop of 0
 Multiplicative Prop of -1
 Pg. 18 #17
 (5) + (-9) =
 (5) – (9) =
 (-5) +(9) =
 (-5) + (-9)=
 (-5) – (-9) =
(+)(+) =
 (-5) – (9)=
(+)(-)=
(-)(-)=
(-)(+)=
(+)/(+)=
(+)/(-) =
(-)/(-) =
(-)/(+) =
 To subtract any number, add the
opposite
(-)(-)(-) =?
(-)^15 =?
(-) ^20 =
What’s the rule?
 Objectives: Solve
equations with one
variable.
 Vocabulary:
 Root = Answer
 Formula = an equation showing a relationship between two
or more variables.
 Constant = Number that doesn’t change in a formula
 Tools and Rules:
One Variable equation answers:
-1 solution (most cases)
-No solutions (“empty” set or
“null” set)
-All values are solution
(Identity)
 Examples:
 (1) -7.5(x – 3) = -30
 (2) 3p – (p – 9) = 2(p + 5)
 (3) 3(2s – 3) = 6(s – 1) – 15
 Objectives: Translating
word phrases into
equations and expressions.
 Tools and Rules:
 Vocabulary:
 Uniform Motion- motion at a constant speed
 Isosceles triangle-two sides, two angles are the same.
 Examples:
 D = RT
 See Examples on pg. 43-45 and Pg. 49-52
 Consecutive numbers:
 5 less than a number
 n, n + 1, n + 2,…
 n, n + 2, n + 4...(for odd/evens)
 Area of a Triangle: A = ½bh
 Area of a square or rectangle: A=bh

 What is the sum of 3 consecutive even numbers if the
middle one is x?
 Objectives: Persevere in
mapping out and solving
word problems.
 Vocabulary:
 Patience: Be patient when solving these problems! Don’t
give up!!!
 Tools and Rules:
Plan for Solving a Word Problem:
1) Read problem carefully. Draw a picture.
2)
3)
4)
5)
Write down given info.
Choose a variable to replace unknowns.
Write an equation using variable.
Solve equation to find variable.
Reread problem to check results.
 Examples:
 Group Activity
 Objectives: Solving
simple and combined
inequalities.
 Vocabulary:
 Conjunction- Inequality with AND. (True = Both True)
 Disjunction- Inequality with OR. (True = One is true)
 Tools and Rules:
 <
 >
 <
 >
 ≠
less than
greater than
less than or equal to
greater than or equal to
NOT equal to
 When you divide or multiply by a -#, FLIP
the inequality sign.
 Examples:
 Next page
 Solve, then graph:
3x + 16 < 7
 Solve, then graph:
-2 < -x + 3 < 4
 Solve, then graph:
c + 7 < 4 OR 7 – c < 1
 Objectives: Solve word
problems using
inequalities
 Vocabulary:
 None
 Tools and Rules:
Pg. 70 in textbook (phrase to translation)
 Examples:
 Next slide…
The Granada Hills football team is ordering sweatshirts for a fundraiser. The team is
charged $5.50 for each sweatshirt and there is a delivery charge of $25 for the order.
If the team sells the sweatshirts for $8 each, how many sweatshirts must be ordered
and sold to produce a profit of at least $80?
 Objectives: Solving and
graphing absolute value
sentences.
 Tools and Rules:
Will do on the board
 Vocabulary:
 Absolute Value= the distance between a number and the
origin (0).
 Examples:
 Next page
 |x| = 1
 |x| > 1
 |x| < 1
1) Solve |3x – 2| = 8
2) Solve |3 – 2t| < 5
3) Solve |2z – 1| + 3 > 8
 Objectives: Solve word
problems using
inequalities
 Vocabulary:
 None
 Tools and Rules:
Pg. 70 in textbook (phrase to translation)
 Examples:
 Next slide…
The Granada Hills football team is ordering sweatshirts for a fundraiser. The team is
charged $5.50 for each sweatshirt and there is a delivery charge of $25 for the order.
If the team sells the sweatshirts for $8 each, how many sweatshirts must be ordered
and sold to produce a profit of at least $80?
 Objectives: Solving and
graphing absolute value
sentences.
 Tools and Rules:
 Vocabulary:
 Absolute Value= the distance between a number and the
origin (0).
 Examples:
 Next page
1) Solve |3x – 2| = 8
2) Solve |3 – 2t| < 5
3) Solve |2z – 1| + 3 > 8
 Objectives: Graphing a
linear equation in two
variables and finding the
slope of a line.
 Vocabulary:
 (Referring to Pg. 107, draw a picture and label each of the
following)




X-axis
Y-axis
Origin
Quadrants I, II, III, IV
 Slope- The steepness of a line; change in y over change in x.
 Tools and Rules:
AX + BY = C (where A and B are not 0) is a
linear equation in two variables. (Standard Form)
One way to graph is find random values of x
and solve for y. (create a table).
 Examples:
 Next slide…
 1) Graph:
 (a) (0, -2)
 (b) (1, -4)
 (c) 2x + y = 5
 (d) x = -3
 2) Find the slope of a line containing the points (3, 5) and (4, 2).
1) Graph:
(a) (0, -2)
(b) (1, -4)
(c) 2x + y = 5
(d) x = -3
2) Find the slope of a
line containing the
points (3, 5) and (4, 2).
m = -3/1 or -3
 y = mx + b (m = slope)
 Write each line in slope-intercept form:
 x–y=2
 y + 3x = 1
 2x + 3y = 5
 Do you notice a pattern?
 What is the “rule” when finding the slope from Standard form?
 The slope of the line Ax + By = C (where B is not 0) is –A/B
 Objectives: Find the
equation of a line given
limited information.
 Vocabulary:
 Y-intercept: where graph crosses the y-axis
 X-intercept: where graph crosses the x-axis
 Tools and Rules:
Point-Slope Form y – y1 = m(x - x1)
Slope-Intercept Form y = mx + b
Standard Form Ax + By = C
Parallel lines = same slope
Perpendicular lines = product of slopes = -1
 Examples:
 Next slide…
1) Give the standard form of the equation that goes through the points (1, 3) and (3, 7).
2) Give the standard form of the equation of the line through point P that is:
(a) Parallel to L
(b) Perpendicular to L
P: (1,2) L: 3x + 2y = 4
(*Hint: first find the slope of L, then use point-slope form to find the line and convert it to
Standard Form)
 Objectives: Solving
systems of linear equations
in two variables.
 Tools and Rules:
 See next slide…
 Vocabulary:
 System of equations- a set of linear equations with the
same two variables.
 Examples:
 Next slide…
 When solving a system of equations, there are three possibilities:
 1) The lines intersect  One solution (a point) (x, y)
 2) The lines never intersect and are parallel  No solution
 3) The lines are the same (“coinciding”)  Infinitely many solutions
But wait,
there’s
more…
 There are two ways to solve systems of equations:
 1) Substitution
 2) Elimination (or, “linear-combination”)
 *In both cases, the goal is to get rid of one of the variables so that we can combine the two
equations into one to solve.
1)
Solve using substitution:
2x + 3y = 13
x–y=9
2) Solve using elimination (linear combination):
3x + 2y = 4
2x – 5y = -29
 Solve the system of equations:
4x - y = 7
3x – 5y = 11
Tell whether you used substitution or elimination and why.
Solve the systems:
1)
2x – y = 7
4x – 2y = 9
2)
3x – 2y = x + 8
y=x–4
 Objectives:Graph single
linear inequalities and
systems of inequalities.
 Vocabulary:
None
 Tools and Rules:
 > or >
shade above the line (or to the right)
 < or < shade below the line (or to the left)
 < or > boundary line will be dotted
 < or > boundary line will be solid
 Examples:
 Next slide…
1) Graph 3x – 2y < 8
2) Graph x – 3 < 0
1) Graph 3x – 2y < 8
2) Graph x – 3 < 0
Solve 3x – 2y < 6
1)
Graph the system:
2x + y > 1
x–y<3
 Graph the system of inequalities:
x > 0;
y < 0;
x + y > -1
 Vocabulary:
Function- A relation between two sets (say D and R) that
assigns to each member of D exactly one member of R.
 Objectives:
Understanding and
graphing functions
Domain = Input values
Range = Output values
Value (of a function) = members of its range
 Tools and Rules:
 Ways to test a function:
 Mapping
 Vertical Line Test (of a graph)
 Function notation:
 f(x) = 3x + 2 (Read as: “f of x is 3x + 2)
 Two limitations of domain:
 When denominator = 0

−#
 Book: f, g, F, G, and phi (see pg. 142) = used to name functions
  tells you the “rule” for x
 Examples:
 Next slide…
1)
f(x) = 8 – ½ x ; D = {-4, -2, 0, 2, 4)
2)
If f(x) =
1
𝑥 2 +5𝑥+6
, what is the domain? (Hint: Think what can x NOT b?)
1)
f(x) = 8 – ½ x ; D = {-4, -2, 0, 2, 4)
R = {6, 7, 8, 9, 10}
2)
If f(x) =
1
𝑥 2 +5𝑥+6
, what is the domain? (Hint: Think what can x NOT b?)
D = {x: x ≠ -2 and x ≠ -3)
 3) Pg. 145 #35
 Vocabulary:
 Objectives: Recognizing
Linear equations have a constant change.
and using linear functions
 Tools and Rules:
 Examples:
 Next slide…
 f(x) = mx + b
 Rate of change m =
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒇(𝒙)
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙
 The table shows a linear function. Fill in the missing information and state what its
rate of change is. Then, plot the points and graph the line.

x
f(x)
0
1
1
4
2
3
 Vocabulary:
 Relation- any set of ordered pairs.
 Objectives: Know the
 Function- a relation in which each input value is linked
difference between a
relation and a function.
 Tools and Rules:
 Ways to test a function:
 Mapping
 Vertical Line Test (of a graph)
to a specific output value.
 Examples:
 Next slide…
Use mapping to test whether the relation is a function. Then, check your work by
graphing and using the vertical line test:
(a) {(2, 1), (1, -2), (1, 2)}
(a) {(3, 2), (1, 4), (5, 2), (7, 6)}
 Objectives: To simplify, add,
and subtract polynomials.
 Vocabulary:
See next page (leave lots of room for this section…
 Examples:
 Next slide…
 Tools and Rules:
 When you subtract polynomials, remember to distribute the
negative sign!!
 Working with a partner, open up to pg. 167 in your textbook.
 Individually, write an example for each of the bold vocabulary words.
 Then, trade with your partner, and write down their example also on your paper.
Constant
Monomial (Term)
Coefficient
Degree of a monomial
Like Terms (like monomials)
Polynomial
Simplified Polynomial (Polynomial in Standard Form)
Degree of a polynomial
 Simplify (Combine like terms and put in Standard Form). Then, give the degree of
the polynomial:
1)
3x – x^2 – 9 + 2x^2 + 2 – 5x
2)
Add -3x^2 – 4x + 9 and 2x^3 – 4x^2 – 5
3)
Subtract 3x^2 + 4x – 5 from 2x^2 – 9x + 7
 Objectives: To simplify, add,
 Vocabulary:
and subtract polynomials.
See Homework or pg. 167
 Examples:
 Tools and Rules:
 Simplify polynomials by combining like terms.
 When you subtract polynomials, remember to distribute
the negative sign!!
 Next slide…
 Simplify (Combine like terms and put in Standard Form). Then, give the degree of
the polynomial:
1)
3x – x^2 – 9 + 2x^2 + 2 – 5x
2)
Add -3x^2 – 4x + 9 and 2x^3 – 4x^2 – 5
3)
Subtract 3x^2 + 4x – 5 from 2x^2 – 9x + 7
 Objectives: To use laws of
exponents to multiply a
polynomial by a monomial.
 Vocabulary:
None
 Examples:
 Tools and Rules:
 (a and b must be real numbers, m and n must be positive integers)
 1. aman = am+n
 2. (ab)m = ambm
 3. (am)n = amn
 (I will write on the
whiteboard.)
 Pg. 172
 Objectives: Use the
distributive property to
multiply polynomials.
 Vocabulary:
binomial- a polynomial that has 2 terms.
trinomial- a polynomial with 3 terms.
 Examples:
 Tools and Rules:
 FOIL: First, Outside, Inside Last
3 special cases:
 (a + b)2 = a2 + 2ab + b2
 (a- b)2 = a2 -2ab + b2
 (a+ b)(a – b) = a2 - b2
 Next slide
Multiply
1)
(2a – 5)(3a + 4)
2)
(3c + 2)2
3)
(4k + 7)(4k – 7)
4)
(3x + 1)(x2 + 2x – 6)
 Objectives: Use the
distributive property to
multiply polynomials.
 Vocabulary:
binomial- a polynomial that has 2 terms.
trinomial- a polynomial with 3 terms.
 Examples:
 Tools and Rules:
 FOIL: First, Outside, Inside Last
3 special cases:
 (a + b)2 = a2 + 2ab + b2
 (a- b)2 = a2 -2ab + b2
 (a+ b)(a – b) = a2 - b2
 Next slide
Multiply
1)
(2a – 5)(3a + 4) = 6a^2 -7a - 20
2)
(3c + 2)2
3)
(4k + 7)(4k – 7)
4)
(3x + 1)(x2 + 2x – 6)
 Objectives: Find the GCF
and LCM of integers and
monomials.
 Vocabulary:
A factor (noun)- a smaller unit of a number or polynomial
found by division.
To Factor (verb)- To write a number or polynomial as the
product of its individual factors.
Prime #- An integer that is only divisible by itself and 1.
 Tools and Rules:
 Prime Factorization: Writing a number as the product of only
prime numbers.
 (Trick: Divide the number by the smallest prime number until you
have only prime numbers).
 GCF-Greatest Common Factor (Hint: Factor is smaller than the
number).
 LCM-Least Common Multiple (Hint: Multiple is larger than the
number).
 Examples:
 Next slide
Find the GCF and the LCM of the following numbers:
20, 28.
With your seat partner, please solve this problem. Be prepared to discuss.
Next, solve 84, 56, 140
 To find the GCF, take the least power of each common prime factor.
 To find the LCM, take the greatest power of each prime factor.
 Example:
 Find the GCF and the LCM of the following numbers: 72, 108, 126.
 GCF = 18
 LCM = 1512
 Find the GCF and the LCM of the following monomials:
 48u2v2 and 60uv3w
 Find the prime factorization of 98.
 Find the GCF and the LCM for 12a2s3
and 8a2r2s
 Find the prime factorization of 98.
 Find the GCF and the LCM for 12a2s3
and 8a2r2s
 If we think of degree-2 polynomials
as rectangles, we can model them
with tiles:
 Each group will get a box or bag of tiles.
 Each bag will have the following pieces (colors will vary):
 Red = Positive
Blue (or Purple) = Negative.
x
x
Area?
= X2
x
1
1
=X
=1
 Objectives: Factor quadratic
Polynomials.
 Vocabulary:
Quadratic Polynomial= ax2 + bx + c (a ≠ 0)
 Tools and Rules:
 Examples:
 Next slide
 T-chart: (see board)
*Use different method if the x2 term has a coefficient of 1 or
another number.
 Factor x2 + 2x -15
 What is the form of our answer going to be?
 What two numbers are we looking for?
 Hint: Look at the second and last terms. (Reverse “FOIL” in your mind).
 To give us -15, we must get a ____ of two numbers.
 To give us 2x, we must get a _____ of the same two numbers.
 Factor x2 – 2x - 8
 Factor 15t2 – 16t + 4.
 Use a different method for factoring when the leading term (one with degree 2) has a
coefficient > 1.
 Practice:
 Factor 2m2 – m - 1
 Objectives: Factor polynomials
using GCF, special products, and
grouping.
 Tools and Rules:
Perfect Square Trinomials:
1) a2 +2ab + b2 = (a + b)2
2) a2 –2ab + b2 = (a - b)2
Difference of Squares:
3) a2 – b2 = (a + b)(a – b)
 Vocabulary:
None
 Examples:
 Next slide
 Type 1: Two variables
 Type 2: Negative degree 2 term
 Pg. 190 Example 6
 Pg. 189 Example 3
 Factor the polynomials:
 1) 3x^3 – 15x^2
Method:
Find GCF
 2) y^2 – 49
Method:
Use difference of two squares.
 3) xy + 5x + 3y + 15
Method:
Use grouping.
 1) 24x^2y – 40xy
 2) x^2 + 14x + 49
 3) 81 – 4a^2
 4) 10q^2 – 5q + 2qt - t
 Objectives: Factor polynomials
using GCF, special products, and
grouping.
 Vocabulary:
None
 Tools and Rules:
Perfect Square Trinomials:
1) a2 +2ab + b2 = (a + b)2
2) a2 –2ab + b2 = (a - b)2
Difference of Squares:
3) a2 – b2 = (a + b)(a – b)
Sum and Difference of Cubes:
a3 + b3 = (a + b)(a2 – ab + b2 )
a3 – b3 = (a - b)(a2 + ab + b2 )
 Examples:
 Next slide
 x^3 – 8
 (x – 8)(x^2 + 8x + 64)
 2g^4 + 54g
 (g + 3)(g^2 – 3g + 3^2)
 t^3 – 64
 (t – 4)(t^2 + 4t + 16)
 a^6 + b^3
 (a^2 + b)(a^4 – a^2b + b^2)
 Objectives: Solve Polynomial
Equations
 Tools and Rules:
 Vocabulary:
(Leave a space, wait for later)
 Examples:
 (Leave a space, wait for
(Leave a space, wait for later)
later)
 Is your polynomial an expression or an equation? Discuss with your partner.
 Set your polynomial equal to zero.
 Now it is known as a polynomial equation. (Write this in the vocabulary section).
 Polynomial equation- A polynomial that equals zero, or its equivalent*.
 *It could be rearranged.
 If we wanted to solve the equation, we would need to find what value of the
variable makes the equation a true statement.
 The roots of a polynomial are the values of x (or variable) that satisfies the equation.
(Write this in vocabulary).
 Factor the equation
 Use the zero-product property*
 Zero-Product Property: ab = 0 if and only if a = 0 or b = 0.
 1st: Write the equation with 0 as one side
 2nd: Factor the polynomial side of the equation.
 3rd: Solve the equation by setting each factor equal to 0.
 * Write in Tools and Rules section
 Solve (x – 5)(x + 2) = 0
 Solution set = {5, -2}
 Solve x^2 = x + 30
 Solution set = {-5, 6}
 Solve 3x^2 = 4x(2x – 1)
 Solution set = {0, 2/3, 2}
 1) q(2q + 6) = 0
 {0, -3}
 2) t^2 + 7t + 6 = 0
 {-1, -6}
 3) 12 + 4m = m^2
 {6, -2}