Transcript Algebra 2 - cloudfront.net

Chapters 1 and 2
Real Numbers
Natural Numbers
 Whole Numbers
 Integers
 Rational Numbers
 Irrational Numbers


Imaginary Numbers

The opposite or additive inverse of any
number a is –a
 The sum of opposites is 0

The reciprocal or multiplicative inverse
of any number a is 1/a
 The product of reciprocals is 1
Properties of Real Numbers
Commutative
 Associative
 Identity
 Inverse
 Distributive

Absolute Value
The absolute value of a number is its
distance from zero on the number line
 | -4| = 4
|0|=0
 | -1 ∙ (-2) | = |2| = 2

Evaluating Algebraic Expressions
When you substitute numbers for the
variables in an expression and follow the
order of operations you evaluate the
expression
 evaluate a – 2b + ab for a = 3 and b = -1
 a – 2b + ab = 3 – 2(-1) + 3(-1)

= 3 – (-2) + (-3)

=3+2–3

=2

Combining Like Terms
A term is a number, variable or the
product of a number and one or more
variables.
 The coefficient is the numerical factor in
a term.
 Like terms have the same variables
raised to the same powers.
 Combine like terms by adding
coefficients

Try these – in your notebooks









Evaluate 7x – 3xy for x = -2, y = 5
16
Evaluate (k-18)2 -4k for k = 6
120
Combine Like Terms
2x2 + 5x – 4x2 + x – x2
-3x2 + 6x
-2(r + s) – (2r + 2s)
-4r – 4s
Practice:

p15 (1-45)odd

Please check you answers in the back
of your book when you are done
1.3 & 1.4 Solving Equations and Inequalities
EQ: What are the steps to solving linear
equations and inequalities?

Warm Up: Solve these problems in your
notebook. (Left hand side)

Simplify each expression
5x – 9x – 3
2y + 7x + y – 1
10h + 12g – 8h – 4g
(x+y)–(x–y)
- (3 – c) – 4(c – 1)





1.3 & 1.4 Solving Equations and Inequalities
EQ: What are the steps to solving linear equations and
inequalities?





5x – 9x – 3 = -4x - 3
2y + 7x + y – 1 = 7x + 3y - 1
10h + 12g – 8h – 4g = 8g + 2h
( x + y ) – ( x – y ) = 2y
- (3 – c) – 4(c – 1) = -3c + 1

Solving Equations – by steps
Distribute
2. Combine Like Terms
3. Combine constants
4. Solve for variable
1.
Solving Equations
A number that makes an equation true is
the solution to the equation.
 Try these:
 8z + 12 = 5z – 21
 z = -11
 6(t – 2)= 2 (9 – 2t)
T=3








Stations:
Pair up - pick an A and a B. You will turn in
ONE sheet of paper with all the problems
solved.
Begin at the station on your table.
Student A does the A problem explaining
each step to Student B
Student B does the B problem explaining
each step to Student A
Add your answers together. They should add
to the number on the equation paper.
Once they do, you may move to the next
station.
Solving For a Variable
Solving for a variable means isolating
that variable on one side of the
equation.
 Solve d = rt for t
 Solve
A = ½ h ( b1 + b2) for h
 Try these:
 Solve P = 2L + 2W for W
 Solve E = ½ mv2 for v

Solving Inequalities
Solve just like equations.
 Reverse the direction of the inequality
symbol if you multiply or divide by a
negative.
 Graph the solution.
 Example: 6 + 5 (2 – x) ≤ 41

Solving Inequalities – Try these
Solve and graph
 3x – 6 < 27
 X < 11
 12 ≤ 2 ( 3n + 1) + 22
 N ≥ -2

Compound Inequalities

A pair of inequalities joined by and or or

3x – 1 > -28 and 2x + 7 < 19
Try this:
 X – 1 < 3 or x + 3 > 8
 2x > x + 6 and x – 7 < 2









Exit Pass: Solve these equations and inequalities on
a sheet of paper. Place in the Algebra 2 basket on
your way out the door.
1. 16x – 15 = -5x + 48
2. 4w – 2(1 - w) = -38
3. -2x < 3 ( x – 5) graph the solution
4. 3x + 4 ≥ 1 and -2x + 7 ≥ 5 graph the solution
Homework:
p21 (1-27) odd
p29 (1-33) odd
1-5 Absolute Value Equations
and Inequalities
EQ: How do you solve equations with absolute value?


1.
2.
3.
4.
5.
6.
7.
warm up
Solve these equations
5(x-6) = 40
5b = 2(3b-8)
2y + 6y = 15 – 2y + 8
4x + 8 > 20
3a – 2 ≥ a + 6
4(t-1) < 3t + 5
.
1-5 Absolute Value Equations and
Inequalities
EQ: How do you solve equations with absolute value?

The absolute value of a number is its
distance from zero on the number line
and distance is non-negative.
Absolute Value Equations
Usually have two solutions
 | 2y – 4 | = 12 means
 2y – 4 = 12 or 2y – 4 = -12
 Isolate the absolute value
 Rewrite as two equations
 Solve both equations
 Be sure to check your answers – they
may not always work.

Try these
| 3x + 2 | = 7
 X = 5/3, -3
 3|4w – 1| - 5 = 10
 W = -13/5, 5
 | 2x + 5 | = 3x + 4
 X = 1,
-9/5 is an extraneous solution

1-5 Absolute Value Inequalities
| 3x + 6 | ≥ 12 - rewrite the equation as:
 3x + 6 ≥ 12 or 3x + 6 ≤ -12

 Note: The inequality symbol changes
direction for the negative solution

3x + 6 ≥ 12 or 3x + 6 ≤ -12

Solve |2x – 3| ˃ 7, graph the solution
1-5 Absolute Value Inequalities
First isolate the absolute value
expression
 3|2x + 6| -9 ˂ 15

1-5 Absolute Value Inequalities
Exit Pass:
1. | x + 3 | = 9
2. |3x – 6| - 7 = 14
3. |6 – 5x| = -18
4. 2 | x + 3 | ≥ 10
5. | 2x + 4 | - 6 < 0

homework
 p 36 1-53 every other odd, except 29
 (1,5,9,13,17,… etc)


Warm up: Complete a 2 minute quick
write in your notebook about how to
solve absolute value equations and
inequalities.
There will be a test next
Tuesday/Wednesday on solving linear
equations and inequalities, including
absolute value problems.
 There will be basic probability questions.

1-6 Probability
EQ: How do you calculate experimental
and theoretical probability?
Probability measures how likely it is for an
event to occur.
 Expressed as a percent- 0% to 100% or
 as a number 0 to 1
 The probability of an impossible event is 0%
 The probability of a certain event is 100%


When you gather data from observations you
can calculate an experimental probability.
The set of all possible outcomes is called the
sample space
 You can calculate theoretical probability as a
ratio of outcomes.

Carnival Fish!
 Homework:
 page 42 (7-21, 25-33)odd
 page 45 (51-61) odd

Warm Up

Glue the warm up slip into your notebook
and complete (page 56)
Stations Review
Fold a sheet of binder paper in half
lengthwise and width wise so there are
four sections on each side.
 You will move from station to station
completing each set of review problems
in a section.
 You answers should add together to get
the number on the station poster.
 Show your work!

2-1 The Coordinate Plane
In an ordered pair ( x,y) the first number
is the x coordinate and the second
number is the y coordinate
 The x-y coordinate plane is divided into
four quadrants by the
x and y axes

2-1 Relations and Functions
A relation is a set of pairs of input and
output values
 The domain is the set of all inputs, or x
values of the ordered pairs
 The range is the set of all outputs, or y
values of the ordered pairs

2-1 Relations and Functions
2-1 Relations and Functions
What is the domain
and range of this
relation?
 Domain {-3, -1, 1}
 Range {-4, -2, 1, 3}

2-1 Relations and Functions
What is the
domain and
range of this
relation?
 D {-2, -1, 1, 3}
 R { -2, 0, 4, 5}

2-1 Relations and Functions
A function is like a
machine. Put an
input (x) in and get
an output (y) out.
 A function is a
relation in which
each element of the
domain is matched
with exactly one
element in the
range.

2-1 Relations and Functions
2-1 Relations and Functions

Vertical line test – If a vertical line
passes through at least two points on a
graph, then the relation is NOT a
function
2-1 Relations and Functions
Function notation
 Y = 2x can be rewritten as
 f(x) = 2x, and read “f of x”
 It does not mean f times x
 To evaluate the function at x = 3 write
 f(3), read “f of 3”

Use the function f(a) = 2a + 3
 Evaluate the function at:
 f(-5)
 f(-3)
 f(1/2)
 f(4)

2-1 Relations and Functions
Homework
 p 50 (3-35) odd: Chapter 1 Test

2-2 Linear Equations
EQ: How do you graph a line in standard
form?
A function whose graph is a line is a
linear equation
 Because the value of y depends on the
value of x, y is called the dependent
variable and x is the independent
variable
 The y intercept is the point where the
line crosses the y axis (x = 0)
 The x intercept is the point where the
line crosses the x axis (y = 0)

2-2 Linear Equations
The standard form of a linear equation is
Ax + By = C and is graphed by finding
the x and y intercepts
 Example: 3x + 2y = 120


Graph 2x + y = 20
2-2 Linear Equations
Slope is the ratio of the vertical change
to the horizontal change
 Slope = vertical change (rise)
horizontal change (run)

Given two points (x1, y1) and (x2, y2)
Slope = y2 – y1
x2 – x1
2-2 Writing Equations
Point-Slope form of an equation
 y – y1 = m ( x – x1)
 Write equation when given a point and
slope
 Ex: Write in standard form an equation
of the line with slope -1/2 through the
point (8, -1)

2-2
Try these
 Write in slope intercept the equation of
the line with slope 2, through the point
(4, -2)
 Write in slope intercept form the
equation of the line with slope 3, through
the point (-1, 5)

2-2
Writing an equation given two points.
 (1,5) and (4, -1)
 (4, -3) and (5, -1)
 (5, 1) and (-4, -3)

2-2
Slope Intercept form
 Y = mx + b
 M is the slope
 B is the y intercept
 To find the slope of a line in standard
form, solve the equation for y

2-2

Find the slope of 4x + 3y = 7
3x + 2y = 1
 3x – 12y = 6

2-2
Parallel lines have the same slope
 Perpendicular lines have slopes that are
opposite reciprocals of each other


The line perpendicular to y = 3x +7 will
have a slope of – 1/3
2-3 Direct Variation
EQ: How do you determine if a function is a direct variation?

Practice:
find the slope between (3,-5) and (1,2)
2. write in slope intercept form the
equation of the line through (-3,-2) and
(1,6)
3. write in standard form the equation of
the line with slope 2, through (-1,3)
1.
2-3 Direct Variation
EQ: How do you determine if a function is a
direct variation?
A linear function y = kx represents direct
variation. The slope k is constant.
 You can write k = y/x, and y/x is the
constant of variation
 The rate of change of the function k is
constant.
 A direct variation function always
contains the point (0,0)

2-3 Direct Variation
EQ: How do you determine if a function is a direct variation?

What does the graph of a direct variation
look like?
2-3 Direct Variation
EQ: How do you determine if a function is a
direct variation?

Direct Variation from a table. k = y/x

For each table, find y/x for each pair of
points.
2-3 Direct Variation
Identify direct variation from an equation
 Must be able to put equation in the form
y = kx
 3y = 2x
 Y = 2x + 3
 Y = x/2
 7x + 4y = 10

2-3 Direct Variation
EQ: How do you determine if a function is a direct variation?

1.
2.
3.
4.
5.
6.
Direct Variation Activity – Rotate for each task
Group chooses direct variation function. Writes an ordered
pair that represents the function on their poster.
Next group determines the constant of variation k for the
given point. (k = y/x)
Next group writes the equation for the direct variation in
the form y=kx.
Next group constructs a table containing 5 other points
that would be on the line.
Next group plots those points and constructs the line
through them.
Final group checks all the work and verifies that all parts
have been done correctly.
2-3 Direct Variation
EQ: How do you determine if a function is a direct variation?

Homework assignment: page 76 (1-45)
odd

Chapter 1 make up test on Wednesday
during enrichment.
2-3 Direct Variation
Can use direct variation to solve some
problems – set up as a proportion
 Suppose y varies directly with x, and x
= 27 when y = -51. Find x when y = -17.

Homework
P 70 (21 -33) odd, (39 – 57) odd
 P 76 (1 – 21) odd

2-4 Using Linear Models
Both equations represent direct
variations
 If y = 4 when x = 3, find y when x = 6


If y = 7 when x = 2, find y when x = 8
2-4 Using Linear Models
EQ: How do you use linear equations to
model real-world situations?
y=mx + b
 m = slope which is a rate of change

 speed, rate of increase or decrease etc

b = a starting value
 beginning height, distance, weight etc
 result = (rate of change) ∙ x + (start value)
2-4 Using Linear Models
Jacksonville, FL has an elevation of 12
feet above sea level. A hot air balloon
taking off from Jacksonville rises 50
ft/min.
 Write an equation to model the balloon’s
elevation as a function of time


result = (rate of change) ∙ x + (start value)
Graph the equation
 Interpret the intercept at which the graph
intersects the vertical axis.

Using two points to make a model
A candle is 6 in. tall after burning for 1 hour.
After 3 hours it is 5 ½ inches tall.
 What is the rate of change? (Slope)
 Write an equation in slope intercept form to
model the height y of the candle after it has
been burning x hours.
 What does the y intercept 6 ¼ represent?

Using models to make predictions
Using the equation for the candle.
 In how many hours will the candle be 4
inches tall?
 How tall will the candle be after burning
for 11 hours?
 When will the candle burn out?


whiteboard problems
Scatter plot
A scatter plot is a graph that relates two
different sets of data by plotting the data
as ordered pairs.
 You can use a scatter plot to determine
a relationship between the data sets.
 A trend line is a line that approximates
the relationship between the data sets in
a scatter plot.

Correlation in a scatter plot
Draw a trend line
that has about the
same number of
points above and
below it
 Use the slope and y
intercepts to
estimate the
equation of the line

Group work

whiteboard problems

page 83 (1-13) all

Draw a graph of 𝑦 = 2𝑥 − 1

Discuss with your neighbor how the
graph of 𝑦 = 2𝑥 − 1 would be different
than the one above. How would it be the
same?

Draw a graph of what you think 𝑦 =
2𝑥 − 1 looks like.
2-5: Absolute Value Functions
and Graphs
Characteristics of Absolute Value
Functions
Absolute value graphs always:
 Have a “V” shape.
 Are symmetric.
 Have straight line sides.
 Take the form 𝑦 = |𝑚𝑥 + 𝑏| + 𝑐
 The point at the bottom (or top) of the V
is the vertex.

2-5: Absolute Value Functions
and Graphs
Graphing Absolute Value Functions

1.
How to graph an absolute value
function:
Find the x coordinate of the vertex by
𝑏
using −
𝑚
Make a table of values that has two
values of x lower than the vertex and
two values above.
3. Plot the points from your table, and
connect them to finish your graph.
2.
2-5: Absolute Value Functions
and Graphs
2-5: Absolute Value Functions
and Graphs
2-5: Absolute Value Functions
and Graphs
Homework: page 92 (1-9, 19-27) odds
 Please make your graphs large enough
to read!

Practice

Graph these absolute value functions

Y = | 3x + 6 |

Y = | x – 1| - 1
2 – 6 Families of functions
EQ: How do translations affect the graph of a parent
function?
A family of functions is made up of
functions with common characteristics
 A parent function is the simplest function
with these characteristics
 A translation shifts a graph horizontally,
vertically or both. It results in a graph of
the same shape and size but possibly in
a different position.

2 – 6 Families of functions
EQ: How do translations affect the graph of a parent function?
Absolute value functions
 y = |x|
parent function
 y = |x| + k shifts vertex of function k units up
(down if negative)
y=|x–h|
shifts vertex of function h units to
the right (to the left if h is negative)
 y = a|x| stretches |x| by a factor of a (slope)
 y = -a|x| reflects the graph of |x| over the x axis

2 – 6 Families of functions
EQ: How do translations affect the graph of a parent function?

y = a|x – h| + k
what does h do?
 what does k do?
 what does a do?

2 – 6 Families of functions
EQ: How do translations affect the graph of a parent function?

How is each graph different from the
parent function y = |x|?
y = |x+1|
 y = -|x|
y=|x|-3
y=|x-2|+4

2 – 6 Families of functions
EQ: How do translations affect the graph of a parent function?

homework: page 99 (1-11, 17-19) all

Chapter 2 test on Monday October 1
2 – 6 Families of functions

Graph y = |x|

On the same graph, graph
y = |x| + 3

On the same graph, graph
y = |x| - 2
Describe how adding a constant outside the
absolute value affects the graph of the parent
function
2 – 6 Families of functions

Explain how a function of the form
y= |x| + k is different from the parent
function.
A vertical translation moves the graph of
the parent function up (or down) k units.
 Write the equation for the graph of
y = |x| translated 5 units down.
 Y = |x| translated 7 units up.

2 – 6 Families of functions
On a new graph, draw the parent
function y = |x|
 On the same graph, draw y = |x + 2|
 On the same graph draw y = | x – 4|


Describe how adding a number inside
the absolute value affects the graph of
the parent function
2 – 6 Families of functions
For a positive number h, y = | x - h| is a
horizontal translation of the parent
function to the right h units
 Y = |x + h| is a horizontal translation h
units to the left.

2 – 6 Families of functions
2 – 6 Families of functions
Graph y = 2 |x|
 Graph y = - |x|
 Graph y = ½ |x|

How does multiplying a graph by a number
larger than one affect the graph?
 How does multiplying a graph by a number
less than one affect the graph?
 How does multiplying by a negative affect the
graph?

2 – 6 Families of functions
A vertical stretch multiplies all y values
by the same factor greater than one,
stretching the graph vertically
 A vertical shrink multiplies all y values by
a factor less than one, compressing the
graph vertically
 Multiplying by a negative factor reflects
the graph over the x axis

2 – 6 Families of functions

A function is a vertical stretch of y = |x|
by 5 – what is the equation?

Reflect the function across the x axis.
What is the equation?
2 – 6 Families of functions
2 – 6 Families of functions
Write equations for the graphs obtained
by translating y = |x|
 10 units right
 4 units down
 7 units left, 6 units up
 Reflection across x axis
 Vertical shrink by a factor of 2/3

Homework
Page 92 (33-43) odd
 Page 99 (1-13) odd
 Page 102 (1-10)


Chapter 1 & 2 test next week Tuesday
Warm up

Graph the following functions
y = 2x + 3
 y = -1/3x +1
y=x–4
x=5

2-7 Two Variable Inequalities
A linear inequality is an inequality in two
variables whose graph is a region in the
coordinate plane that is bounded by a
line.
 To graph a linear inequality:
 Graph the boundary line
 Determine which side of the line
contains solutions
 Determine if the boundary line is
included

2-7 Two Variable Inequalities
A dashed boundary line indicates the
line is not part of the solution
 A solid boundary line indicates the line is
part of the solution
 Choose a test point to check if a region
makes the inequality true – use (0,0), if it
is not on the line
 Example: graph y > ½ x - 1

2-7 Two Variable Inequalities
Try this on your whiteboard – graph:
 y ≤ 2x + 3

Graph the line y = 2x + 3
 Check the test point (0,0)
 Is the line part of the solution?

2-7 Two Variable Inequalities

Graph y ˃ -4x + 3
2-7 Two Variable Inequalities
Graph the absolute value inequality
y≤|x–4|+5


-y + 3 > | x + 1 |
Homework:
 Page 106 (1,5,9,11,15,19, 25)
 Corrections to quiz – use quiz as study
guide


Chapter 2 test on Monday
Study Guide Answers
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
a
b
c
d
b
a
b
d
c
a
c
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
d
b
b
d
a
a
b
b
d
c
a
23)a
24)c
25)a
26)c
27)c
28)a
29)c
30)c
31)c
32)d
33)b