algebra ii - canyon crest academy library media...

Algebra II Chapter 2 Notes with Section 1-7

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

1

5. Solve: 6. Solve:

Solving Absolute Value InequalitiesWhat 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

2

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 TimeSlow 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 TimeSlow Train

Fast Train

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HW #5: Pg. 55 #3-69 x 3; Motion Problem worksheet #1-2

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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.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7

-6-5

-4-3-2-1

123

4567

89

10y

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.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7

-6-5

-4-3-2-1

123

4567

89

10y



Wider, narrower, or the same width as : ____________

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3.) Graph the absolute value function by completing the x-y table below:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7

-6-5

-4-3-2-1

123

4567

89

10y




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.)




5.)




6

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 :

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

Compare the location of the vertex and the width of this graph to that of .

7.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y


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8.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y


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.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

9.) 10.)

8

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8

-7-6-5-4-3-2-1

12345678910

y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910

y


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 TimeSlow 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

9

Slow train

Fast train

HW#6: Pg. 127: #3-33 x3 Motion Problem worksheet: #4-5

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Notes#7: Section 2.1 Relations and functions, Section 2.2: Slope and rate of changeRelation: 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 10 0

Use the vertical line test to determine whether each relations is a function:4.) 5.)

11

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

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6.)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

7.)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

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

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

Is this relation a function? Why or why not?

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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 points3.) Connect the points

13.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

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14.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

Slope: The slope of a line describes its _______________ and its _______________

Sketch:Lines with a positive

slopeLines with a negative

slopeLines with undefined

slopeLines 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 =

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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 = -3 28.) (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.

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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 = 3 32.) 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)


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 TimeSD 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

18

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 term is positive

1.) Convert to slope-intercept form:

4x – 12y = 8

2.) Convert to standard form:

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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 points on 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.)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8

-7-6-5

-4-3-2-1

123

45

6789

10y

(Graph for #4 and #5)

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m = ______

b = ______


7.) x + 3y = -6

m = ______

b = ______


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

(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 different points

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- 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 points and graphing as just one

10.) x + 2y = 4 x -intercept y -intercept (set y = 0) (set x = 0)

x-int: ( , 0) y-int: (0, )

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4

-3-2-1

12

345678910

y

11.) 3x – y = 3

x-int: ( , ) y-int: ( , )-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

12.) 2x – 3y = 8

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

123456789

10y

Special Cases: Graphing Horizontal and Vertical Lines

• Any line in the form x = ___ is a _______________ line because it intersects the ___ _________

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• 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.

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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.

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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 toy = 3x – 5. Leave your answer in standard form.

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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

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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 < 61.) (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:

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9.)

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-10

-9-8-7

-6-5-4

-3-2-1

123

456

789

10y

10.)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8-7

-6-5-4

-3-2-1

123

456

789

10y

11.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8-7

-6-5-4

-3-2-1

123

456

789

10y

12.) Graph:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8-7

-6-5-4

-3-2-1

123

456

789

10y

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 6Y 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?

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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)

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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.

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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!!

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algebra ii - canyon crest academy library media...

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