Algebra II Chapter 2 Notes with Section 1-7

Algebra II Chapter 2 Notes with Section 1-7

Notes# 5: Sec. 1.7 – Solve absolute value equations and inequalities

What does absolute value mean?

Recall that the absolute value of a number is its ______from zero on a number line. Since absolute value represents ______, it can never be ______.

What does solving an absolute value equation mean?

means to find the places on the number line that are ______away from ______.

Solution: ______Graph:

means to find the places on the number line that are ______away from ______.

Solution: ______

Solving Absolute Value Equations:

Get the | | alone
Write two equations; one ______and one ______(never change what is inside the absolute value bars!)
Solve for x; expect ______answers
Check both answers by ______

1. Solve and check:

(checks): / 2. Solve and check:

(checks):

3. Solve and check:

/ 4. Solve and check:

5. Solve:

/ 6. Solve:

Solving Absolute Value Inequalities
What do absolute value inequalities mean?
means “What numbers are ______2 units away from zero?” Graph the solution: Is this an “AND” or an “OR” graph?

means “What numbers are ______2 units away from zero?” Graph the solution: Is this an “AND” or an “OR” graph?
Solving Absolute Value Inequalities:

Get | | alone on the left side of the inequality
Write 2 equations, one ______

and one ______(SWITCH THE INEQUALITY SIGN!)

If use ______

If , use ______

Graph and solve for x - sometimes, put back in sandwich: (small number) < x < (big number)

7. Solve and graph:

______
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 / 8. Solve and graph: > 3.

______
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
9. Solve and graph:

______
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 / 10. Solve and graph: < 2

______
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
11. Solve and graph:

______
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 / 12. Solve and graph:

______
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Motion Problems: use a chart!

D = R x T
Same direction ____, Opposite direction ____, Round trip _____

13. Two cars start at the same place but travel in opposite directions. One car averages 42 miles per hour and the other averages 51 miles per hour. How many hours will it be before the cars are 651 miles apart?

Distance / Rate / Time
Slow Car / / /
Fast Car / /

14. Two trains leave the station at the same time going in opposite directions. One of them travels 40mph and the other travels at 50mph. In how many hours will they be 135 miles apart?

Distance / Rate / Time
Slow Train / / /
Fast Train / /

HW #5: Pg. 55 #3-69 x 3; Motion Problem worksheet #1-2
Notes #6: Sections 2.7 – Graphing Absolute Value Functions, Transformations

1.) Graph the absolute value function by completing the x-y table below. Then find the information requested.

/

Shape of graph: _____, Direction of opening: _____

Vertex: ______, Slope of right side of graph: _____

2.) Graph the absolute value function by completing the x-y table below. Then find the information requested.

/
Shape of graph: _____, Direction of opening: _____

Vertex: ______, Slope of right side of graph: _____
Wider, narrower, or the same
width as : ______
3.) Graph the absolute value function by completing the x-y table below:

/
Shape of graph: _____, Direction of opening: _____

Vertex: ______, Slope of right side of graph: _____
Wider, narrower, or the same
width as : ______
Let’s generalize: Absolute value function graphs
Standard form:
Shape: ______
Direction of opening: ______when a is positive ______when a is negative
Vertex: ______
Slope of right side of graph: ______
Width of graph compared to : ______when
______when
______when
Without graphing, find the requested information:
4.)

Shape of graph: _____, Direction of opening: _____

Vertex: ______, Slope of right side of graph: _____
Wider, narrower, or the same
width as : ______/ 5.)

Shape of graph: _____, Direction of opening: _____

Vertex: ______, Slope of right side of graph: _____
Wider, narrower, or the same
width as : ______

Graphing Absolute Value Functions:

Identify the vertex (h, k) and plot that point
Choose two x-values to the left and two x-values to the right of the vertex. Plug into your x-y chart and plot these points. Your points should make a V-shape.
Compare with: How is it translated? How does its width compare?

6.) Graph :
Compare the location of the vertex and the width of this graph to that of .

7.) Graph:
Compare the location of the vertex and the width of this graph to that of .

8.) Graph:
Compare the location of the vertex and the width of this graph to that of .

Transformations: You will describe the stretching and/or translation of an image on a coordinate plane using function notation.

is shown. Use this graph to sketch the graph of the given function:

For each of the points listed on the graph, follow these steps:
1.) x-coordinate: ______

2.) y-coordinate: ______

3.) multiply by: ______
Now graph your three new points and connect. /
9.)

/ 10.)

Motion Problems: use a chart!

D = R x T
Same direction ____, Opposite direction ____, Round trip _____

11.) Two trains travel for 11 hours, starting from the same place traveling in opposite directions. One train travels at an average rate that is 11 miles per hour faster than the other one. Find the rate of each train if they are 635 miles apart.

Distance / Rate / Time
Slow train / / /
Fast train

12.) Two passenger trains started at the same time from towns 810 miles apart and met in 6 hours. The rate of one train was 5 miles per hour slower than that of the other. Find the rate of each train.

Distance / Rate / Time
Slow train / /
Fast train

HW#6: Pg. 127:#3-33 x3

Motion Problem worksheet: #4-5

Notes#7: Section 2.1 Relations and functions, Section 2.2: Slope and rate of change

Relation: any set of ______(or ______)

Domain: the set of ______of a relation written in { }. The domain values are often called the ______.

Range: the set of ______of a relation written in { }. The domain values are often called the ______.

Function: a relation in which each _____ is paired with a unique _____. Meaning, in a function, the ______- coordinate should not be repeated.

Vertical Line Test:
- If any vertical line touches the relation more than once then it is ______a function
- If all vertical lines touch the relation zero or one time, then it ______a function

Examples:

Find the domain and range of each relation. Is the relation a function? Why or why not?

1.) {(3, 2), (7, 0), (2, 1), (3, 4)} 2.) Domain Range 3.)

x / y
-2 / 6
3 / 8
-1 / 1
0 / 0

Use the vertical line test to determine whether each relations is a function:

4.)
/ 5.)

6.)
/ 7.)

8.) In this relation of ordered pairs: (-2, -3), (-1, 1), (1, 3), (2, -2), and ( 3,1)
What is the domain? / What is the range?

Represent the relation by mapping
Domain Range

/ Graph the relation

Is this relation a function? Why or why not?

Linear function: when graphed, a linear function forms a non-vertical ______
A linear function is written in the form:

Tell whether the function is linear or not. Then find the value of the function at the indicated x-value.
9.) Find

/ 10.) Find

11.) Find

/ 12.) Find

Graphing an equation with two variables / 1.)make a T chart with 5 values for x. Plug in the x values and solve for y
2.)Plot the points
3.) Connect the points
13.) Graph:

/
14.) Graph:

/

Slope: The slope of a line describes its ______and its ______

Sketch:

Lines with a positive slope / Lines with a negative slope / Lines with undefined slope / Lines with zero slope

Finding slope

Find two points on the line. Your job is to get from the left point to the right point.
Count how many steps up you need to go (this is the rise)
Count how many steps across you need to go(this is the run)

Find slope of the line. (Double check for +/- )

15.) m = /
16.) m = /
17.) m =
18.) m = / 19.) m = / 20.) m =
Without using a graph and given two points:
and
Slope = m = /

Use the formula to find the slope of the lines containing these points.

Hint: first label points as and

21.) (2, 2) (8, 9)22.)(-4, -6) (3, -2)23.) (-2, 3) (2, 1)

24.) (5, -11) (-9, 4)25.)(9, 7) (3, 7)26.) (4, -6) (4, 0)

Each pair of points lies on a line with the given slope. Find x or y.

27.) (3, 1), (x, 7) slope = -328.) (4, y), (2, 8) slope =

Find the slopes of the graphed lines below:

Parallel Lines:

/ Perpendicular Lines:

Parallel lines have ______slopes. Perpendicular lines have ______, ______slopes.

The slope of a line is given. Find the slope of a line parallel to it and a line perpendicular to it.

29.) 30.)

31.) m = 332.) m = 0

Tell whether the lines are parallel, perpendicular or neither:
33.) Line 1: through ( -2, 2) and ( 0, -1)
Line 2: through (-4, -1) and ( 2, 3) / 34.) Line 1 : through ( 1, 2) and ( 4, -3)
Line 2 : Through ( -4, 3) and (-1, -2)

35.) Line 1: through ( -4, -2) and ( 1, 7)
Line 2: through ( -1, -4) and ) 3, 5)

Motion Problems: use a chart!

D = R x T
Same direction ____, Opposite direction ____, Round trip _____

36.) Maya drove her car from San Diego to Los Angeles at an average rate of 50 miles per hour and returned over rainy roads averaging only 20 miles per hour. Find the time going and returning if the time returning was 2 hours more than the time going.

Distance / Rate / Time
SD to LA
/ / /
LA to SD
/

37.) Two cars leave town at the same time heading in the same direction. One car travels at 60mph and the other travels at 40mph. After how many hours will they be 50 miles apart?

Distance / Rate / Time
Slow car / / /
Fast car /

HW#7: Pg. 76: 3-39 x3, Pg. 86: 3-21x3, 29-34 all, Motion Problem Wkst: 3, 6

Notes#8: Section 2.3: Graphing lines,Section 2.4: Writing equations of lines

Forms of Lines: Lines can be written in either Slope-Intercept form (y = mx + b) or

Standard Form (Ax + By = C). You need to know how to convert from one to the other.

Converting to Slope-Intercept Form
Goal: y = mx + b
(where m and b are integers or fractions)

Get y alone
Reduce all fractions

/ Converting to Standard Form
Goal: Ax + By = C
(where A, B, and C are integers and
where A is positive)

Get x and y terms on the left side and the constant term on the right side of the equation
Multiply ALL terms by the common denominator to eliminate the fractions

If necessary, change ALL signs so that the x termis positive

1.) Convert to slope-intercept form:
4x – 12y = 8

/ 2.) Convert to standard form:

3.) Convert to both slope-intercept form and standard form:
a) y – 3 = -5(x + 4) b.)

Section 2.3: A. Graphing Lines using the slope and y-intercept:
- Get y alone so the equation is in y = mx + b form (m = ______, b = ______)
- Graph b first. This point goes on the ____ axis.
- Use slope and count rise over run to the next point(s). When you have at least three pointson your graph,connect the points with a ruler to make a straight line.
- Label your graphed line with the original equation
Most common errors:

Graphing b on the x-axis instead of the y-axis
Graphing the slope in the wrong direction (e.g. forgetting a negative)

4.)
(I’m already in slope-intercept form!)

m = ___ ( graph me second!)

b = ___ ( graph me first! I go on the y-axis!)
5.)
(I’m already in slope-intercept form!)

m = ___ ( graph me second! Watch the negative!)

b = ___ ( graph me first! I go on the y-axis!)
Compare the graph to

6.) x – 2y =2
(Get me in slope-intercept form first. Get y alone.)

m = ______

b = ______
Compare the graph to

7.) x + 3y = -6

m = ______

b = ______
Compare the graph to

/
(Graph for #4 and #5)

(Graph for #6 and #7)

B. Graphing Equations Using Intercepts

x-intercept is the x-coordinate of the point where a line crosses the ______. To find the x-intercept, make y = 0 and solve for x.
y-intercept is the y-coordinate of the point where a line crosses the ______. To find the y-intercept, make x = 0 and solve for y.
Remember: the intercepts are TWO different points!

Find the x- and y-intercepts.

8.) 9.)

x-intercept y-intercept

(make y = 0) (make x = 0)

(____, 0) and (0, ____)

Graphing Lines using the x- and y- intercepts. The intercepts are the point(s) where a line intersects the axes of the coordinate plane.
- Find the x and y intercepts (by setting the opposite variable to zero)
- Write these answers as two differentpoints
- Graph and connect these points to graph the line
- Label the graphed line with the original equation
Most common error:
Forgetting that the intercepts are two different pointsand graphing as just one
10.) x + 2y = 4
x-intercept y-intercept
(set y = 0) (set x = 0)

x-int: ( , 0) y-int: (0, ) /
11.) 3x – y = 3

x-int: ( , ) y-int: ( , ) /
12.) 2x – 3y = 8
/
Special Cases: Graphing Horizontal and Vertical Lines

• Any line in the form x = ___ is a ______line because it intersects the ______

• Any line in the form y = ___ is a ______line because it intersects the ______
Use this pattern to graph these lines without a table of solutions.
13.) y = 3 / 14.) x = -2 / 15.) y = -4

Writing Equations of Lines:

A. Writing linear equations given the slope and y-intercept

- Find the slope (m) and y-intercept (b) [If the given information is a graph, then you will have to count by hand to find these values.]

- Fill in m and b so you have an equation of the line in y = mx + b form. Convert to standard form if necessary. y = ______x + ______

(Put m here!) (Put b here!)

16.) Find the equation of the line with slope of 5 and y-intercept of -2. Write in standard form.

/ 17.) Find the equation of the given line in slope-intercept form.
/ 18.) Write the equation of a line that has the same slope as and has a y-intercept of 1. Write in standard form.

B. Writing linear equations given the slope and a point

plug slope = m into y = mx + b
name your point (x, y) and plug these values in for x and y
solve for b
plug m and b back into y = mx + b **(leave x and y as variables!**

convert to standard form, if necessary

19.) Find the equation of the line with slope of -2 and going through (-1, 3) in slope-intercept form.

/ 20.) Find the equation of the line with slope of and going through (6, -2) in standard form.

C. Writing linear equations given two points

find the slope using the slope formula (m = ______)
pick one of your points to be x and y

plug m, x, y into y = mx + b
solve for b; plug m and b into y = mx + b (** Remember to leave x and y as variables! **)

convert to standard form, if necessary

21.) Find the equation of the line going through (-3, 1) and (4, 8) in slope-intercept form.

/ 22.) Find the equation of the line with
x-intercept 3 and y-intercept -2 in standard form.

23.) Find the equation of the line going through (5, 2) and (-1, 3) in standard form.

/ 24.) Find the equation of the line with
x-intercept 5 and y-intercept -4 in slope-intercept form.

Writing an equation of a line given a parallel or perpendicular line

Find m from the given line by getting y alone first
If the line is parallel, ______
If the line is perpendicular, ______
Plug m, x, y into y = mx + b; solve for b
Plug m and b into y = mx + b (leave x and y as variables!). Convert to standard form if necessary.

25.) Write the equation of the line that passes through (1, 2) and is parallel to y = 3x + 4. Leave in slope-intercept form.

/ 26.) Write the equation of the line that passes through (3, -2) and is perpendicular to
y = 3x – 5. Leave your answer in standard form.

27.) Write the equation of the line that passes through (2, 4) and is parallel to 2x – y = 5. Convert to standard form.

/ 28.) Write the equation of the line that passes through (-1, 5) and is perpendicular to
2x – 3y = 6. Leave in slope-intercept form.

29.) Write the equation of the line that passes through (-3, 2) and parallel to x + 2y = 4. Leave in slope-intercept form.

/ 30.) Write the equation of the line that passes through (-2, 0) and perpendicular to x = 2y – 1. Leave in standard form.

HW#8: Pg. 93: 3-54 x 3, Pg. 101: 3-45 x 3

Notes#9: Section 2.8: Graph linear inequalities in two variables; Review for Test

Tell whether the given ordered pair is a solution of 5x – 2y < 6

1.) (0, -4) 2.) ( 2, 2) 3.) (-3, 8) 4.) (-1, -7)

Graphing Linear Inequalities:

Get y alone and into y = mx + b form
Graph this boundary line using m and b
2 decisions:

Line:
If connect points with a ______
If connect points with a ______/ Shading: If shade ______the line
If shade ______the line
OR Test a point not on the line (use (0, 0) if possible) to see if it is a solution to the inequality. If it is a solution then shade side of line containing that point. If it is not a solution then shade other side of line.

5.) Graph: 6.) Graph: x < 2

7.) Graph: y > -2x 8.) Graph:

9.)

/ 10.)

11.) Graph:
/ 12.) Graph:

Chapter 2 Review Problems! Please do the following problems on a separate sheet of paper.

For questions 1-4, use the table below.

X / 5 / 8 / 0 / -2 / 5 / -4 / 6
Y / 9 / 2 / -4 / -1 / 0 / 6 / 1

1. Graph the relation.

2. What is the domain of the relation?

3. What is the range of the relation?

4. Is the relation a function?

For questions 5-7, find the slope of each line, using the slope formula, passing through the two given points. Then tell whether each line is rising, falling, horizontal, or vertical.

5. (4, 8) and (-7, 2)6. (6, -3) and (-2, -3)7.

For questions 8-9, graph each equation.

8. 9.

For questions 10-12, write the equation of each line described below in (a) slope intercept form and (b) standard form.

10. passes through (8, 2) and (6, -3)

11. perpendicular to the line 4x – 2y = 9 and passes through (6, -1)

12. has an x-intercept of -5 and a y-intercept of 9

13. parallel to the line 5x + 3y = 9 and passes through (7, -3)

For questions 14-15, graph each inequality.

14. 15.

For questions 16-17, use .

16. 17.

For questions 18-19, find the vertex of the graph of the following functions, determine whether the graph of the function opens up or down, state whether the graph is wider, narrower, or the same width as the graph of y = |x|. Then graph the function.

18. 19.

For question 20, write the equation of the function graphed below.

20.

HW# 9: Pg.135 # 3-27 x3, Pg. 145 #1-13 all, 20-24 even,

Chapter 2 & Section 1.7 Test and Notes Check tomorrow!!