· Web viewNJ Center for Teaching and Learning ~ 21 ~ . Alg II: Linear, Exp ... a system of linear...

Algebra-Geometry I GRAPHING

SUCCESS CRITERIA:

1) Understand the key components of the coordinate plane.

2) Understand how to graph a line given different pieces of information.

3) Understand how to graph compound inequalities.

4) Understand how to graph inequalities on a coordinate plane.

5) Understand how to graph an exponential equation.

I

INSTRUCTOR: Craig Sherman Hidden Lake High School

Westminster Public Schools

PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org

EMPOWER Recorded TARGET SCALE THEME

MA.09.EE.03.04 Graph Equations & Inequalities

MA.09.F.01.04 Graphing Functions denoting Key Features

PROFICIENCY SCALE:

SCORE REQUIREMENTS

4.0 In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class.

Score 4.0 does not equate to more work but rather a higher level of performance.

3.5 In addition to Score 3.0 performance, in-depth inferences and applications with partial success.o Determine the appropriate solution in a Real World example.

3.0 The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught.

o Solve linear equations, ANDo Solve linear inequalities ANDo Solve a system of two linear equations, ANDo Solve a system of two linear equalities, ANDo Translate real world problem into a system of linear equations or inequalities.

2.0 Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as the

learner…o Understand the key parts of a coordinate grah, ORo Understand how to graph a point, ORo Understand how to graph using a table, OR o Understand how to graph using the x and y intercepts, OR o Understand how to find the slope given two points, ORo Understand how to graph given a point and slope, OR o Determine if a line is horizontal or vertical, OR o Determine if two lines are perpendicular, ORo Determine if two lines are parallel, ORo Be able to write an equation from given information, ORo Be able to graph compound inequalities, ORo Be able to graph inequalities on a coordinate plane, ORo Be able to graph exponential equations.

1.0 Know and use the vocabulary Identify the Basic ElementsWith help, a partial understanding of some of the simpler details and process

PMI-NJ Center for Teaching & Learning ~2~ NJCTL.org

Graph using a table

WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

table

ordered pair

axis

axis label

axis scale

slope

rate of change

INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA

NJ Center for Teaching and Learning ~ 3 ~ www.njctl.org

https://www.sophia.org/tutorials/32-graphing-linear-equations-using-an-x-y-chart

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-solutions-to-two-var-linear-equations/v/graphs-of-linear-equations

Classwork

For the equations below, make a table with at least 3 ordered pairs, plot the points and connect them to form the line.

1. y = 3x – 42. y = -2x + 43. y = x – 3

4. y = 12x + 4

5. y = - 23x + 1

6. Make a table for the rule y = 3x – 1. Then use the table to graph the rule neatly. Be sure to include scales and labels for the axes and the graph.

Homework

For the equations below, make a table with at least 3 ordered pairs, plot the points and connect them to form the line.

7. y = -x – 2 8. y = 2x + 1

9. y = 14 x

10. y = -2x – 2

11. y = - 13x + 4

12. Make a complete graph for y = 3x + 113. Using the rule y = –2x + 5,

a. Make a table for the rule.b. Use your table to graph your rule on graph paper. Make sure your graph is complete.


Graph using the slope and y-intercept


y-intercept

x-intercept

zeros



https://www.sophia.org/tutorials/graphing-a-linear-equation-using-slope-and-the-y-i

https://www.khanacademy.org/math/algebra-basics/core-algebra-graphing-lines-slope/core-algebra-graphing-slope-intercept/v/graphing-a-line-in-slope-intercept-form

Classwork

14. Use lines A, B, C and D to answers the questions.

a. What is the y-intercept of each line?b. Is the slope of each line positive, negative, zero or undefined?

15. Use the lines to answer the following questions.

a.

a. What is the slope of lines E, F, G and H?b. What are the equations of lines E, F, G, and H?


D

C

BA

E

FG

H

Homework16. Use lines I, J, K, and L to answers the questions.

a. What is the y-intercept of each line?b. Is the slope of each line positive, negative, zero or undefined?

17. Use lines M, N, O, and P to answers the questions.

a. What are the slopes of lines M, N, O, and P?b. What are the equations of lines M, N, O, and P?


I J

K

L

M

PON

18. Match each clue (a) through (d) below with the line on the graph that fits the description. Then write a clue for the line that has no match.

a. I have a large negative value for m and a small positive value for b.

b. I have a small negative value for m and a value of zero for b.

c. I have a small negative value for m and a large positive value for b.

d. I have a small positive value for m and a small positive value for b.


Line 2

Line 1

Line 4

Line 5Line 3

Slope Formula


Classwork

19. Find the slope of the line through each of the following two points.

a. (-12,-5), (0,-8)

b. (12,-18),(11,12)

c. (-18,-20),(-18,-15)

d. (-20,-4),(-12,-10)

e. (8,10),(0,14)

f. (6,9),(3,-9)

g. (1,2),(5,7)

h. (3,-3),(12,-2)

i. (-4,-8),(-1,1)

j. (4,7),(-3,7)


https://www.sophia.org/tutorials/finding-the-slope-when-given-two-points

https://www.khanacademy.org/math/algebra/two-var-linear-equations/slope/v/introduction-to-slope

Homework

20. Find the slope of the line through each of the following two points.

a. (3,-9),(1,1)

b. (7,4),(3,8)

c. (-3,0),(5,12)

d. (8,-2),(12,-2)

e. (6,-3),(2,9)

f. (-3,7),(-4,8)

g. (5,9),(5,-8)

h. (-5, 0.5),(-6,3)

i. (-7,1),(7,8)

j. (-2,1),(5,7)

21. Growth equations for different types of plants (x = time in days and y = height in inches) are listed on

the left, and descriptions for their growth patterns are listed on the right. Match the equations with their

descriptions. State a reason for each match that you make. Then write an equation for the plant

description left over.

a.

b.

c.

d.

1) Plant # 1 starts as a seed and grows at a rate

of 4 inches every 2 days. It takes an extra day to

emerge from the ground.

2) Plant # 2 was already 1 inch tall when planted

and grows half of a foot every 4 days.

3) Plant # 3 started as a seed and grows 6

inches every 2 days.

4) Plant # 4 was already 4 inches tall when

planted and grows 1 foot every 12 days.

5) Plant # 5 was already 1 inch tall when

planted and grows 8 inches every 2 days.


Graph using point - slope


point-slope


Classwork

22. Graph each equation

a. y – 2 = (x – 3)

b. y + 5 = 2(x + 2)

c. y – 4 = -3(x – 3)

d. y + 7 = 12(x – 1)

e. y – 3 = - 34 (x + 5)

Homework

23. Graph each equation

a. y – 2 = (x + 4)

b. y + 5 = -2(x – 4)

c. y – 1 = 4(x – 2)

d. y – 5 = - 13 (x – 1)


https://www.sophia.org/concepts/graphing-a-linear-equation-in-point-slope-form

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-slope-intercept-form/v/equation-of-a-line-1

e. y – 0 = 15 (x – 3)

Graph using interceptsINSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA

Classwork

24. Rewrite the equations in standard form. Graph the intercepts and then the line that passes through them.

a. y = 3x + 4

b. y = -2x + 3

c. y = x + 7

d. y = 25x – 2

e. y = - 3x

f. y = - 23x + 5

g. y – 4 = 2(x – 5)

h. y + 5 = -3(x – 4)

i. y + 6 = 14 (x – 6)

j. y – 3 = (x + 7)

Homework25. Rewrite the equations in standard form. Graph the intercepts and then the line that passes through

them.a. y = 6x + 4

b. y = -3x – 2c. y =

16x – 3


https://www.sophia.org/tutorials/graphing-using-intercepts

https://www.khanacademy.org/math/algebra-basics/core-algebra-graphing-lines-slope/core-algebra-graphing-intercepts/v/graphing-using-x-and-y-intercepts

d. y = x + 4

e. y = - 27x – 2

f. y = -7x

g. y – 5 = 2(x – 3)

h. y + 2 = -4(x – 2)

i. y – 3 = 37 (x – 3)

j. y + 1 = 2(x + 1)

Horizontal & Vertical Lines


horizontal line

vertical line

Classwork

26. Determine if the following equations are horizontal, vertical, neither or cannot be determined.


a. y = -5b. x = 7c. 2x + 4y = 8

d. 7x – 21 = 0e. 3x + 2y = 3x – 4

Homework

27. Determine if the following equations are horizontal, vertical, neither or cannot be determined.

a. x = -5b. y = 7c. 8x + -4y = -2

d. -6x – 3y = -6x + 2e. -8x = -24

Parallel and Perpendicular Lines


parallel line

perpendicular line

NOTES:


Classwork

28. Match the following parallel equations from each column.

a. y = 3x – 2

b. y = -2x + 1

c. y = 13x – 7

d. 6x – 2 = 7y

e. 2(y + x) = 18

I. 3y = x + 18

II. -4x + 17 = 2y

III. 6y + x = 0

IV. -5(3y – 9x) = 30

V. 14y = 12x + 28

29. Match the following perpendicular equations from each column.

a. y = -4x – 4

b. 2y = 3x + 12

c. 7x + y = -10

d. y – 5 = 12(x + 4)

e. 3(4y + 8x) = 0

I. y + 3 = -2(x – 5)

II. 4y = x – 4

III. 2x = 14y -8

IV. y + 4 = 12(x – 4)

V. 2x + 3y = 15

30. Write an equation that would be parallel to the line y = 4x + 7

31. Write an equation that would be perpendicular to the line 4x + 6y = -12


Homework

32. Match the following parallel equations from each column.

a. y = 3x+ 7

b. 4y = 16x + 12

c. 6x + 12y = 1

d. y – 4 = 45 (x – 11)

e. 0 = 22x – 11y

I. 24x – 6y = 18

II. y + 5 = 2(x – 13)

III. 2y + x = 24

IV. 3(15y – 12x) = 27

V. -12x = -4y – 16

33. Match the following perpendicular equations from each column.

a. 6y = -12x + 9

b. y – 4 = 15 (x + 7)

c. 5(2x – 8y) = 35

d. 35x – 5y = 0

e. 19 + y = -3x

I. y – 9 = -4(x – 2)

II. 3x + 21y = 0

III. 8(14y – 7x) = 24

IV. y = -5x + 5

V. -10x + 30y + 20 = 0

34. Write an equation that would be parallel to the line 6y = -24y – 18

35. Write an equation that would be perpendicular to the line y – 4 = 25 (x + 4)


36. Find the equation of the line parallel to the line that goes through the point (10, 8). Show

all work.

37. Examine line segments A and B at right. Find the slope of each segment. Explain whether the segments are perpendicular by analyzing the slopes.

38. Line W is parallel to Line X and perpendicular to Line Y.

Line Y passes through the origin and intersects Line W at the point . Line X has an x-intercept of 5. Find the equations of all three lines. Explain your reasoning.


B

A

Write an equation that would be parallel to the line 6y = -24y – 18

39. Write an equation that would be perpendicular to the line y – 4 = 25 (x + 4)

40. Find the equation of the line parallel to the line that goes through the point (10, 8). Show

all work.

41. Examine line segments A and B at right. Find the slope of each segment. Explain whether the segments are perpendicular by analyzing the slopes.

42. Line W is parallel to Line X and perpendicular to Line Y.

Line Y passes through the origin and intersects Line W at the point . Line X has an x-intercept of 5. Find the equations of all three lines. Explain your reasoning.


B

A

How to write equations from given information

Classwork

43. Write an equation in point slope form for the line through the given point with the given slope.

a. (3,4); m = 6

b. (-2,-7); m = - 32

c. (7,-4); m = -3

d. (4,0); m = 1

e. (-4,-4); m = 13

44. A line passes through the given points. First write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form.

a. (-1,0), (1,2)b. (3,5), (0,0)c. (6,-2), (9,-8)

d. (-1,-5), (-7,-6)e. (-3,4), (3,-2)

45. Write the equation of the line through the given points in standard form.

a. (6,3), (-4, 2)b. (12,3), (-8,-4)c. (15,-4), (8, -3)

d. (7,2), (8,5)e. (20,-10), (-30, 0)

46. A line has an x-intercept of 8 and y-intercept of 12.a. Write an equation for the line.b. Write an equation that is parallel to this line.c. Write an equation that is perpendicular to the line from part a.d.

47. Write the equation of the line through (-3,-2) and is parallel to the line y = -2x + 5

48. Write the equation of the line through (7,4) and perpendicular to y = 12x – 5

49. Find another point on the line containing the points (–2, –3) and (3, – 4). Write the coordinates of the point you found.

50. Consider the line containing the points (–1, 3) and (4, 6).

a. Find the slope of the line. b. Find the y-intercept.


c. Find another point on the line that has integer coordinates. Write the coordinates of the point you found.

d. Write an equation for the line.

Homework

51. The points (2, 3) and (6, 11) lie on a line. Show your work when answering the following questions.

a. What is the growth rate of this line?

b. What is the y-intercept?

c. What is the equation of the line?

52. Write an equation in point slope form for the line through the given point with the given slope.

a. (5,-4); m = -2b. (-2, -3); m = 4 c. (7,4); m =

14

d. (9,-9); m = 3

e. (6,0); m = - 23

53. A line passes through the given points. First write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form.

a. (-10,-50), (5,25)b. (0,10), (10,-20)

c. (0.5,9), (50,-90)d. (8,9), (5,-6)

e. (-7, 4), (-3,6)

54. Write the equation of the line through the given points in standard form.

a. (1,4), (-1,1)b. (5,-3), (3,4)

c. (2,4), (-3,-6)d. (5,3), (4,5)

e. (0,0), (-1,-2)

55. A line has an x-intercept of -4 and y-intercept of 16.a. Write an equation for the line.b. Write an equation that is parallel to this line.c. Write an equation that is perpendicular to the line from part a.

56. Write the equation of the line through (8,5) and is parallel to the line y = x + 7

57. Write the equation of the line through (-2,5) and perpendicular to y = - 23x + 3

58. Without graphing, find the equation of the line that passes through the points (– 4, 3) and (6, –2). Show all work in a clear and organized manner.

59. It costs $20 plus $1.50 per hour to rent a golf cart.


a. Find an equation showing the relationship between the cost of renting a cart (y) and the number of hours it was rented (x).

b. Draw a graph to show this relationship.

c. Show how to use your graph to estimate the cost to rent a cart for 5 hours.

Compound Inequalities & Special Cases


compound inequality


Classwork60. Write the compound inequality for the graph

below.

61. Write the compound inequality for the graph below.


https://www.sophia.org/tutorials/compound-inequalities-how-to-write-graph-and-use-t

https://www.khanacademy.org/math/algebra/one-variable-linear-inequalities/compound-inequalities/v/compound-inequalities



64. Solve each inequality and graph the solution on a number line.

a. 7 ≤ x + 4 < 10b. 4 < x + 6< -2c. 15 ≤ x – 3 ≤ 9d. 3x > -9 and 8 < 2xe. x + 3 > 5 and -4x ≥

-8

f. 3x < 5 and 2x – 2 ≤ -3

g. -3x > -12 and 2x > 6

h. x + 3 > 15 or x – 7 < -3

i. 4x > -12 or 2x > 12

j. 2x + 4 > 12 or 3x + 12 < 15

k. 5x + 7 < 12 or -7x > -21

l. 4x ≤ 20 or 3x + 1 > -5

65. Model each situation with an inequality and solve.a. Your friend is on a diet. He is supposed to eat at least 1500 but no more than 1800 calories per day.

Before dinner he has had 1050 calories. What number of calories should he eat at dinner?b. To get an A in class your total points must be between 690 and 750, inclusive. If you have 652 points

before the last test. What possible numbers of points could you earn and still get an A in the class.

Homework






70. Solve each inequality and graph the solution on a number line.

a. -4 < x + 2 < 9b. 7 ≤ 3 – 2x < 11c. 4 ≤ x + 3 < 9d. x ≥ 0 and x < 25e. 8x > -64 and 8 > 4xf. -2 > 3x + 7 and 3x + 1 ≤ -14

g. x + 3 < 12 and 4x > 24h. -9x – 3 < 6 or 3x + 5 ≥ 26i. 6x – 8 > 10 or -9x > -9j. 10x ≤ -20 or 20x ≥ 40k. -4x + 3 > -17 or 4x – 8 ≥ 4l. -7x + 10 < 31 or 10x + 20 ≤ -50

71. Model each situation with an inequality and solve.a. The sum of any two sides of a triangle is greater than the length of the third side. If two sides of a triangle

are 5 inches and 17 inches. Find the range of values for the lengths of the third side.b. A gymnast wants her average score after 3 events to be between 9.0 and 10. If she scores an 8.7 and

9.1, what possible values for the third event will make her average between 9.0 and 10?

Graphing Linear Inequalities


half plane


Classwork


https://www.sophia.org/tutorials/graphing-linear-inequalities-8

https://www.khanacademy.org/math/algebra-basics/core-algebra-graphing-lines-slope/core-algebra-graphing-linear-inequalities/v/graphing-inequalities

Graph each linear inequality.

72. 2y – 4x < -273. y < 10x + 474. y > 8x75. y > -½x – 476. -3y < -9x + 3

77. 6x – 4y < -1278. 16x + 6y > 36

79. y >

65 x – 8

Write the inequality described then graph the inequality.

80. Suppose your class is raising money for their school trip. You make $5 on each smoothie and $3 on each soft pretzel that you sell. How many items of each type must you sell to raise more than $150?

a. Write a linear inequality that describes the situation.b. Graph the linear inequality.c. Write two possible solutions to the problem.

Homework

Graph each linear inequality.

81. y > -482. x + 4y < 883. -5x + 4y > -24

84. y <

35 x + 5

85. 2x + 3y < -1586. y > -2x + 7

87. y <

49 x – 6

88. –x + 3y < –12

Write the inequality described then graph the inequality.

89. Suppose that you exercise by either running 5 miles per day or rollerblading 10 miles per day. How many days will it take you to cover a distance of at least 150 miles?

a. Write a linear inequality that describes the situation.b. Graph the linear inequality.c. Write two possible solutions to the problem.


Graphing Exponential FunctionsWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

horizontal asymptote



https://www.sophia.org/concepts/the-graph-of-an-exponential-equation

https://www.khanacademy.org/math/algebra/introduction-to-exponential-functions/graphs-of-basic-exponential-functions/v/graphing-exponential-functions

Class WorkGraph each equation. Make sure that the y-intercept and the horizontal asymptotes are clear. Please number the axes on your graphs.

90. y=4 (3)x 91. y=0.4(2)x 92. y=(0.3)x+2

93. y=3 (.45)x−2 94. y=30 (0.2)x+15 95. y=12(3)−x


97. y=6(0.4)−x+2

HomeworkGraph each equation. Make sure that the y-intercept and the horizontal asymptotes are clear.Please number the axes on your graphs.

98. y=4 (0.3)x 99. y=4 ( 4 )−x 100.y=3 (0.4)x+5

101. y=3 (5)x−8 102. y=12(0.5)−x+30 103. y=10 (5)x


104. y=2(.45)x+3

GRAPH: Linear Equations, Inequalities, & Exponentials Unit Review

Multiple Choice– Choose the correct answer for each question. No partial credit will be given.

1. Which value of x is in the solution set of the inequality 7(x + 1) > -14?

a. -22b. -2c. -1d. 1

2. Which graph represents the solution set for: 1/5 + 1/3 x ≤ 4/15

a.

b.

c.

d.

3. Which type of line is used when graphing -4x - 3y ≤ 10

a. Dashed


b. Solid

c. No line

For #4 and #5, Determine which point is a solution to each linear inequality.

4. 4x - 2y ≤ 6

a. (7, 9)

b. (0, -4)

c. (-1, -8)

d. (-10, -100)

5. -5x - 3y > 10

a. (-2, 0)

b. (0, -2)

c. (-3, -1)

d. (1, -4 )

For #6 and #7, Choose the linear inequality that describes each graph.

6. a. 4y < -3x +12

b. 3x – 4y > 12

c. y > ¾ x + 3

d. 3x + 4y > 12

7.a. y ≥ -10x + 11

b. 10x + 2y ≤ 11

c. y ≥ -5x + 5.5

d. 10x – 2y ≤ 11


8. The equation that models exponential decay passing through (0,5) and a horizontal asymptote of y = 4 is

a. f ( x )=5 ex+4b. f ( x )=−1e x+4

c. f ( x )=5 e−x+4d. f ( x )=−1e− x+4

9. A forest fire spreads continuously at a burning 10% more acres an hour. How long will it take for 1000 acres to be on fire after 200 acres are burning?

VI. 23.026 hoursVII. 16.094 hours

VIII. 6.932 hoursIX. not enough information

Short Constructed Response – Write the correct answer for each question. No partial credit will be given.

10. Graph: 3x +2 < 0

12. Graph: 5x – 4 ≥ 6 and 7x + 11 < 32

13. Graph: 22 ≥ 2(2m – 1) or 5 – 3m ≤ -13

14. Graph: 7x + 4 < 18

15. Graph: 3x + y > 1

16. Graph: y < -1/9x - 1

19. Write the inequality for the graph below.




22. Most snakes live where the temperatures ranges from at least 75°F to no more than 90°F. Graph the temperatures where snakes will not thrive.

Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.

23. Model with an inequality and solve. Graph your solution.

Maria is in charge of organizing the school’s holiday carnival. She is setting up booths in the gym. Each booth can have three players. Maria plans on setting up at least 22 booths in the gymnasium. What is the minimum number of players that can compete in the booths at the holiday carnival?

24. The fast food restaurant in town is selling hot dogs and hamburgers to raise money for a local charity. They are selling the hot dogs for $2.00 each and the hamburgers for $3.50 each. What is the minimum number of hot dogs and hamburgers the restaurant can sell in order to raise at least $1000 for the charity?

a. Write an inequality that describes the situation.

b. Graph the inequality.


c. Write three possible solutions to the problem.

d. Why is “200 hot dogs and 200 hamburgers” not a solution?


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