The document is about algebra and graphing. It contains 7 lessons: negative numbers, finding points on a grid, graphing ordered pairs, problem-solving strategies using logical reasoning, functions, graphing functions, and a problem-solving investigation. Each lesson contains examples and practice problems to teach the concepts and standards covered in that lesson.
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. Chapter 12
Algebra and Graphing
Click the mouse or press the space bar to continue.
2. Algebra and Graphing
12
Lesson 12-1 Negative Numbers
Lesson 12-2 Find Points on a Grid
Lesson 12-3 Graph Ordered Pairs
Lesson 12-4 Problem-Solving Strategy: Use
Logical Reasoning
Lesson 12-5 Functions
Lesson 12-6 Graph Functions
Lesson 12-7 Problem-Solving Investigation:
Choose a Strategy
3. 12-1 Negative Numbers
Five-Minute Check (over Chapter 11)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
Example 3
4. 12-1 Negative Numbers
• I will understand and use negative numbers.
• positive number
• negative number
5. 12-1 Negative Numbers
Standard 4NS1.8 Use concepts of
negative numbers (e.g., on a number line, in
counting, in temperature, and in “owing”).
6. 12-1 Negative Numbers
Write the number that represents the situation.
Then show the number on a number line. Sara
owes her mom $5.
When you owe money, it is a decrease.
The number is –5.
7. 12-1 Negative Numbers
Write the number that represents the situation.
The temperature is 7 below zero.
A. –7
B. 7
C. 0
D. –8
8. 12-1 Negative Numbers
Write the number that represents the situation.
Then show the number on a number line. Grant
earned $7 for shoveling Mr. Lincoln’s driveway.
When you earn money, it is an increase.
The number is 7 or +7.
9. 12-1 Negative Numbers
Write the number that represents the situation.
Julia deposits $25 into her bank account.
A. 0
B. 25
C. –25
D. 30
10. 12-1 Negative Numbers
Write the number of each letter on the number line.
A is between –4 and –2. So, A is –3.
B is between –2 and 0. So, B is –1.
C is the same distance from zero as +4.
Answer: So, C is +4.
11. 12-1 Negative Numbers
Write the number of each letter on the number line.
A. –5; –4; –3
B. –3; –1; 4
C. –3; –2; –1
D. –3; 1; 4
12.
13. 12-2 Find Points on a Grid
Five-Minute Check (over Lesson 12-1)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
14. 12-2 Find Points on a Grid
• I will use ordered pairs to find and name points on
a grid.
• coordinate plane • y-axis
• origin • ordered pair
• x-axis • coordinates
15. 12-2 Find Points on a Grid
Preparation for Standard 4MG2.1 Draw the
points corresponding to linear relationships on
graph paper (e.g., draw 10 points on the graph of
the equation y = 3x and connect them by using a
straight line).
16. 12-2 Find Points on a Grid
What is
located at
point (2, 5)?
To find (2, 5), start at
(0, 0). Move right 2
units. Then, move up
5 units.
Answer: The ordered pair (2, 5) locates the post office.
17. 12-2 Find Points on a Grid
What is located at
point (1, 3)?
A. red house
B. blue house
C. tan house
18. 12-2 Find Points on a Grid
What letter is located
at (–3, 5)?
To find (–3, 5), start at
(0, 0). The –3 tells you
to move 3 units to the
left. The 5 tells you to
move 5 units up.
Answer: The ordered pair (–3, 5) locates the letter C.
19. 12-2 Find Points on a Grid
What letter is located
at (–4, 2)?
A. Q
B. R
C. S
20.
21. 12-3 Graph Ordered Pairs
Five-Minute Check (over Lesson 12-2)
Main Idea and Vocabulary
California Standards
Key Concept: Length of Line Segments
Example 1
Example 2
22. 12-3 Graph Ordered Pairs
• I will graph ordered pairs and find the lengths
of line segments on a coordinate grid.
• graph
23. 12-3 Graph Ordered Pairs
Standard 4MG2.2 Understand that the
length of a horizontal line segment equals
the difference of the x-coordinates.
Standard 4MG2.3 Understand that the
length of a vertical line segment equals the
difference of the y-coordinates.
25. 12-3 Graph Ordered Pairs
P
Graph point P at (–2, 4).
Step 1 Start at (0, 0).
Step 2 The x-coordinate
is –2. So, move 2
units to the left.
Step 3 The y-coordinate
is 4. So, move 4
units up.
Step 4 Graph a point at (–2, 4). Label it P.
26. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
A. B.
27. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
C. D.
28. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
C.
29. 12-3 Graph Ordered Pairs
Find the distance
between (2, 2) and
(2, 6).
The line segment is
vertical. Subtract the y-
coordinates.
length of segment
=6–2
=4
Answer: The length is 4 units.
30. 12-3 Graph Ordered Pairs
Find the distance between (1, 5) and (1, 2).
A. 2 units
B. 3 units
C. 4 units
D. 5 units
31.
32. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Five-Minute Check (over Lesson 12-3)
Main Idea
California Standards
Example 1: Problem-Solving Strategy
33. 12-4 Problem-Solving Strategy: Use Logical Reasoning
• I will solve problems using logical reasoning.
34. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Standard 4MR1.1 Analyze problems by
identifying relationships, distinguishing
relevant from irrelevant information,
sequencing and prioritizing information,
and observing patterns.
35. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Standard 4SDAP1.1 Formulate survey questions;
systematically collect and represent data on a
number line; and coordinate graphs, tables, and
charts.
36. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Bella, Devan, Carl, and Jill live on Ash, Pine, Maple,
and Oak Streets. Bella lives on Ash. Devan does
not live on Pine. Carl lives on Maple. What street
does Jill live on?
37. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Understand
What facts do you know?
• The four students live on Ash, Pine,
Maple, and Oak Streets.
• Bella lives on Ash Street.
• Devan does not live on Pine Street.
• Carl lives on Maple Street.
What do you need to find?
• Find what street Jill lives on.
38. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Plan
You can use logical reasoning and a table to solve
the problem.
39. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Solve
• Bella lives on Ash and Carl lives on Maple. So Jill
cannot live on either of these streets.
• Devan does not live on Pine Street. He must live
on Oak Street.
Answer: So, Jill must live on Pine Street.
40. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Check
Look back at the problem. The answer makes sense
for the facts given in the problem.
So, the answer is correct.
41.
42. 12-5 Functions
Five-Minute Check (over Lesson 12-4)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
Example 3
43. 12-5 Functions
• I will complete function tables.
• function
44. 12-5 Functions
Standard 4AF1.5 Understand that an
equation, such as y = 3x + 5, is a prescription
for determining a second number when a first
number is given.
45. 12-5 Functions
Jake makes a $2 profit for each magazine
subscription he sells. Complete the function
table to show the profits earned for the number
of subscriptions sold.
Make a table to find how
much money Jake will
make for the number of
subscriptions sold. 4 2 8
Multiply each input by 2 6 2 12
to find each output.
46. 12-5 Functions
Anita makes $5 for every yard she rakes leaves for.
Complete the function table to show the profits
earned for the number of yards raked.
A. 4 × 2 = 8; 6 × 2 = 12
B. 5 × 4 = 25; 5 × 6 = 35
C. 5 × 4 = 20; 5 × 6 = 30
D. 4 × 4 = 16; 6 × 6 = 36
47. 12-5 Functions
Use the rule y = 4x to complete a function table
where the input is 4, 6, 8, 10.
The rule y = 4x
means to multiply x
by 4 to get y.
48. 12-5 Functions
Use the rule y = 5x to complete a function table
where the input is 2, 4, 6, 8. What is the output for
each input?
A. 10, 20, 30, 40
B. 5, 10, 15, 20
C. 7, 9, 11, 13
D. 2, 4, 6, 8
49. 12-5 Functions
Use the rule y = 2x + 5 to complete a function table
where the input is 3, 5, 7.
First, multiply x by 2.
Then, add 5 to the
product to get y.
50. 12-5 Functions
Use the rule y = 4x + 1 to complete a function table
where the input is 2, 4, 6. What is the output for
each input?
A. 8, 16, 24
B. 9, 17, 25
C. 2, 4, 6
D. 3, 5, 7
51.
52. 12-6 Graph Functions
Five-Minute Check (over Lesson 12-5)
Main Idea
California Standards
Example 1
Example 2
Graph Functions
53. 12-6 Graph Functions
• I will graph functions.
54. 12-6 Graph Functions
Standard 4MG2.1 Draw the points
corresponding to linear relationships on graph
paper (e.g., draw 10 points on the graph of the
equation y = 3x and connect them by using a
straight line).
55. 12-6 Graph Functions
Each smoothie at the bakery costs $3. The function
table shows how much it will cost if you buy
1, 2, 3, or 4 smoothies. Write the ordered pairs and
graph the function y = 3x. Then use the graph to
find how much 6 smoothies will cost.
Step 1 Write the
ordered pairs.
Then graph.
56. 12-6 Graph Functions
Step 2 Extend the
pattern in the
graph by drawing
a straight line.
The straight line
will help you see
the pattern.
Answer: So, 6 smoothies
will cost $18.
57. 12-6 Graph Functions
Noah gets $11 a week for allowance. Find ordered
pairs and graph y = 11x to find how much money
Noah will have after 8 weeks.
A. $70
B. $75
C. $80
D. $88
58. 12-6 Graph Functions
Graph 10 points on the graph of the function
y = 3x – 1.
Complete a table to
find the ordered pairs.
59. 12-6 Graph Functions
Then graph the ordered
pairs on a coordinate
plane. Connect the
points with a straight line.
60. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
A.
61. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
B.
62. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
C.
63. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
D.
64. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
D.
65.
66. 12-7 Problem-Solving Investigation: Choose a Strategy
Five-Minute Check (over Lesson 12-6)
Main Idea
California Standards
Example 1: Problem-Solving Investigation
67. 12-7 Problem-Solving Investigation: Choose a Strategy
• I will solve problems by choosing the best
strategy.
68. 12-7 Problem-Solving Investigation: Choose a Strategy
Standard 4MR1.1 Analyze problems by
identifying relationships, distinguishing
relevant from irrelevant
information, sequencing and prioritizing
information, and observing patterns.
69. 12-7 Problem-Solving Investigation: Choose a Strategy
Standard 4AF1.5 Understand that an
equation such as y = 3x + 5 is a prescription
for determining a second number when a
first number is given.
70. 12-7 Problem-Solving Investigation: Choose a Strategy
AIDEN: I just got a new video game
system. Games cost $20. Felice has
the older version of the video game
system. Her games cost $15. How
many video games can we each buy if
we each have $60?
YOUR MISSION: Find out how many
games each person can buy.
71. 12-7 Problem-Solving Investigation: Choose a Strategy
Understand
What facts do you know?
• Games for Aiden’s game system cost $20.
• Games for Felice’s game system cost $15.
• Each has $60 to spend on video games.
What do you need to find?
• Find how many games each person can buy.
72. 12-7 Problem-Solving Investigation: Choose a Strategy
Plan
To find the answer, organize the data to show the
number of games and the total amount of money
spent.
73. 12-7 Problem-Solving Investigation: Choose a Strategy
Solve
Answer: Since Aiden’s games cost more, he can
only buy 3, whereas Felice can buy 4.
74. 12-7 Problem-Solving Investigation: Choose a Strategy
Check
Look back at the problem. Since 20 × 3 = 60 and
15 × 4 = 60, you know that the answer is correct.
78. Algebra and Graphing
12
(over Chapter 11)
Find the area of the figure.
4
A. 60 6
6
B. 36 4
C. 56
D. 48
79. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter A on the number line.
A. –2
B. 3
C. –3
D. –1
80. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter B on the number line.
A. 3
B. 1
C. –2
D. 2
81. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter C on the number line.
A. –4
B. 4
C. 2
D. 3
82. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point A.
A. (2, –2)
B. (–1, 2)
C. (–2, 2)
D. (–2, –2)
83. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point B.
A. (–5, –1)
B. (5, 1)
C. (–5, 2)
D. (–5, 1)
84. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point C.
A. (–4, 3)
B. (4, –3)
C. (–4, –3)
D. (4, 3)
85. Algebra and Graphing
12
(over Lesson 12-3)
Find the length of the horizontal or vertical line
segment formed by the following set of ordered
pairs: (5, 4), (1, 4).
A. 6
B. 9
C. 4
D. 5
86. Algebra and Graphing
12
(over Lesson 12-3)
Find the length of the horizontal or vertical line
segment formed by the following set of ordered
pairs: (–3, 6), (–3, 0).
A. 3
B. 6
C. 9
D. 0
87. Algebra and Graphing
12
(over Lesson 12-4)
Hugo has horses and ducks on his farm. He has 3
times as many horses as he does ducks. Together,
the animals have 14 legs. How many horses and
ducks does Hugo have?
A. 1 horse, 3 ducks
B. 2 horses, 3 ducks
C. 1 horse, 5 ducks
D. 3 horses, 1 duck
88. Algebra and Graphing
12
(over Lesson 12-5)
Use the rule to complete the function table.
A. 5, 6, 7
B. 9, 18, 27
C. 6, 9, 12
D. 5, 8, 11
89. Algebra and Graphing
12
(over Lesson 12-6)
Choose the set of points that belong on the graph of
the function y = 2x.
A. (1, 2), (3, 6), (5, 8)
B. (1, 3), (2, 4), (3, 5)
C. (2, 1), (4, 2), (6, 3)
D. (1, 2), (2, 4), (3, 6)
90. Algebra and Graphing
12
(over Lesson 12-6)
Choose the set of points that belong on the graph of
the function 2x = y.
A. (20, 10), (18, 9), (16, 8)
B. (10, 12), (9, 11), (8, 10)
C. (10, 20), (9, 18), (8, 16)
D. (10, 20), (18, 9), (8, 16)