MGU Mathematics Series — Arithmetic & Pre-Algebra Foundations
Part IV: Pre-Algebra Foundations
Coordinate Plane · Ordered Pairs · Patterns · Graphing
Chapter 12 of 14
MGU Mathematics Series  ·  Arithmetic & Pre-Algebra Foundations  ·  Library Textbook Edition

Chapter 12:
Coordinate Plane
and Patterns

The coordinate plane, ordered pairs, quadrants, tables of values, linear patterns, pattern rules, and graphing — the visual language of mathematics that every subsequent algebra course requires

■ Math Level: Pre-Algebra ■

■ 14 Sections  •  Part IV of 5 ■

Follows Chapters 1–11 of Arithmetic & Pre-Algebra Foundations  ·  Prepares students for Algebra I linear equations and functions

Chapter 12 — Front Matter

Purpose and Overview

Mathematics is not only calculation. It is also communication, and one of the most powerful ways mathematics communicates is through pictures. A graph turns a list of numbers into a shape. It turns a pattern into a line. It turns a relationship between two quantities into something a person can see, trace, and interpret at a glance.

Chapter 12 introduces the coordinate plane, the standard mathematical system for locating and graphing points in two dimensions. Students will learn how the plane is organized, how to name and plot locations using ordered pairs, how to recognize patterns in tables, and how to represent those patterns visually. By the end of this chapter, students will understand that graphing is not decoration added to mathematics. It is a method of reasoning, a tool for finding relationships, and a foundation for the linear equations and functions that come next in Algebra I.

Opening Question

A weather station records the temperature every hour throughout a single day. At midnight the temperature is 48 degrees. By noon it has risen to 74 degrees. By midnight again it has dropped back to 51 degrees. The data can be written in a table. But what if someone wants to see how the temperature changed, when it rose fastest, and when it began to fall? A table of numbers does not answer those questions easily. A graph does. How do mathematicians turn a list of numbers into a picture, and what does that picture reveal that the numbers alone do not?

14Sections
14Objectives
16Key Terms
12+Worked Examples

Why This Chapter Matters

Every graph in science, business, economics, medicine, and engineering rests on the same foundation: the coordinate plane. Students who understand ordered pairs, axes, and plotting will be prepared to read stock charts, understand scientific data, navigate digital maps, program computers, and study linear equations in Algebra I. More immediately, the coordinate plane ties together the arithmetic and pre-algebra skills from earlier chapters. Negative numbers locate points to the left or below zero. Fractions and decimals appear as precise coordinates. Patterns connect arithmetic sequences to the beginning of algebraic functions. This chapter is the visual language of mathematics. Learning it here means that when it reappears in algebra, geometry, physics, and statistics, students will recognize it as familiar ground.

Learning Objectives

  1. 1Explain why graphs are used to represent numerical relationships.
  2. 2Identify and label the parts of the coordinate plane.
  3. 3Distinguish between the x-axis and y-axis and explain their roles.
  4. 4Write and interpret ordered pairs correctly.
  5. 5Plot points correctly in all four quadrants and on the axes.
  6. 6Name the quadrant or axis location of a point from its coordinates.
  7. 7Explain how the sign of each coordinate determines direction.
  8. 8Create tables of values from arithmetic patterns and rules.
  9. 9Identify and describe simple linear patterns using constant differences.
  10. 10Write a pattern rule in the form y = mx + b for a table of values.
  11. 11Graph a simple linear pattern on the coordinate plane.
  12. 12Read and interpret information from a graph.
  13. 13Recognize and correct common coordinate mistakes.
  14. 14Connect coordinate plane skills to linear equations in Algebra I.
Chapter 12 — Contents

Sections at a Glance

Sections 12.1 – 12.7
The Coordinate Plane: Location, Axes, and Direction
Why Graphs Matter · The Plane · x-Axis and y-Axis · Ordered Pairs · Plotting · Quadrants · Direction
Section 12.1

Why Graphs Matter

Numbers carry information, but they do not always reveal it. A list of values tells you what happened. A graph often shows you how it happened, why it happened, and what is likely to happen next.

Consider two situations. In the first, a student earns $8 per hour working at a store. In one hour she earns $8. In two hours she earns $16. In three hours she earns $24. In four hours she earns $32. In the second, the same numbers are graphed, placing hours along one direction and earnings along the other. Each pair of values becomes a point. When all the points are marked, they fall in a perfectly straight line rising from left to right.

The table answers a specific question: how much for four hours? The graph reveals something deeper: the earnings grow at a constant, unchanging rate. The relationship is linear. That straight line communicates a whole relationship in one image.

Why Visual Representation Is Powerful

Graphs do several things that tables and equations do not do as easily. They show trends: a rising line means growth, a falling line means decrease, a flat line means no change. They show speed of change: a steep line means fast change, a shallow line means slow change. They show comparisons: two lines on the same graph reveal which quantity is larger, where they are equal, and when one overtakes the other. They also reveal patterns instantly, often before a student can write an algebraic equation.

From Data to Graph

The process of making a graph always follows the same logic: collect pairs of related values, place them on a plane, and mark each pair as a point. To do this consistently and correctly, mathematicians use a structured system called the coordinate plane, which this chapter develops section by section.

MGU Library Connection
  • The MGU Reference Tables appendix includes a blank coordinate plane grid suitable for practice work.
  • The Algebra I chapter on linear equations builds directly on the graphing skills introduced here.
Practice 12.1Concept Check
  1. In your own words, explain the difference between a table of values and a graph. Give one thing a graph shows that a table does not make immediately visible.
  2. A student records the number of pages read per day for a week: 20, 25, 25, 30, 28, 35, 40. Would a graph of this data show a rising line, a falling line, or something in between? Explain.
  3. Describe a real-life situation in which a graph would be more useful than a table for understanding a relationship. Explain what the axes would represent.
Section 12.2

The Coordinate Plane

The coordinate plane is a flat surface divided into a grid by two perpendicular number lines. The word "coordinate" means a value that locates a position. The word "plane" means a flat, two-dimensional surface that extends in all directions without ending.

The coordinate plane makes it possible to describe any point on that surface using exactly two numbers. Those two numbers work together: one locates the point horizontally, and one locates it vertically. Together they form a unique address for every location on the plane.

How the Plane Is Built

The coordinate plane is constructed from two number lines placed at right angles to each other, crossing at the zero mark of each line. This crossing point is called the origin. The horizontal number line runs left and right. Positive values go to the right, negative values go to the left. The vertical number line runs up and down. Positive values go upward, negative values go downward. Together these two lines divide the plane into four regions called quadrants, discussed in Section 12.6.

y | Q II | Q I | (neg x, pos y) | (pos x, pos y) | ———————————————————+—————————————————— x | (neg x, neg y) | (pos x, neg y) | Q III | Q IV |
Why Two Dimensions

A single number line locates positions in one direction only. The coordinate plane uses two directions, so it can represent two quantities at once — making it possible to graph a relationship between two things, such as time and earnings, hours and temperature, or input and output in a function.

Student Note — The Cartesian Plane

The coordinate plane is sometimes called the Cartesian plane, named after the French mathematician René Descartes, who developed the coordinate system in the seventeenth century. You will see both names used in future mathematics courses.

Practice 12.2Concept Check
  1. What is the origin? What are its coordinates?
  2. How many numbers are needed to identify a unique location on the coordinate plane? Why does one number not suffice?
  3. A single number line can show one quantity, such as temperature. Give an example of a relationship that would require the coordinate plane rather than a single number line to represent properly.
Section 12.3

The x-Axis and y-Axis

The two number lines that form the coordinate plane each have a name. The x-axis is the horizontal number line. It runs left and right across the plane. Positive x-values are to the right of the origin, and negative x-values are to the left. The y-axis is the vertical number line. It runs up and down through the plane. Positive y-values are above the origin, and negative y-values are below. The two axes are perpendicular — they cross at exactly a right angle — and they meet at the origin, which carries coordinates (0, 0).

What the Axes Represent

In a blank coordinate plane, the axes are simply reference lines for location. When a student graphs a real-world relationship, the axes are usually labeled to show what each direction represents. For example, a graph of time versus distance might label the x-axis "Hours" and the y-axis "Miles." A graph of temperature over a day might label the x-axis "Hour of Day" and the y-axis "Temperature (°F)." In abstract mathematics problems, the axes are simply called the x-axis and y-axis.

Scale on the Axes

The axes are marked with evenly spaced numbers, just like a ruler. The spacing between marks can vary depending on the data being graphed. If values range from 0 to 100, numbering by tens is a reasonable scale. If values range from 0 to 10, numbering by ones is clearer. The scale must remain consistent: if one tick mark represents two units, every tick mark on that axis must represent two units.

Student Warning — Scale Consistency

Always keep the scale consistent on each axis. If one tick mark represents two units, every tick mark on that axis must represent two units. An inconsistent scale distorts the graph and produces misleading pictures.

Practice 12.3Skill Practice
  1. Name the horizontal axis and the vertical axis of the coordinate plane.
  2. A student is graphing data that ranges from −50 to 150 on the y-axis. What scale increment (such as 5, 10, 25, or 50) would you recommend? Explain your reasoning.
  3. A graph has the x-axis labeled "Weeks" and the y-axis labeled "Dollars Saved." In plain language, what relationship does the graph represent?
  4. A student marks 0, 5, 10, 20, 30 on the y-axis at equal spacing. Explain the error and describe the correct labeling.
Section 12.4

Ordered Pairs

A location on the coordinate plane is named by an ordered pair. An ordered pair is written as two numbers separated by a comma and enclosed in parentheses: (x, y). The first number is the x-coordinate, describing horizontal position measured from the origin. The second number is the y-coordinate, describing vertical position. The order matters — the x-coordinate always comes first, the y-coordinate always comes second.

Reading an Ordered Pair

Ordered PairHorizontal MovementVertical Movement
(4, 3)4 units right3 units up
(−2, 5)2 units left5 units up
(3, −4)3 units right4 units down
(−1, −6)1 unit left6 units down
(0, 0)No movementNo movement (origin)
(6, 0)6 units rightNo movement (on x-axis)
(0, −3)No movement (on y-axis)3 units down
Student Warning — Order Matters

The pair (4, 3) and the pair (3, 4) name two different points. Swapping the coordinates changes the location. The x-coordinate always comes first. A very common mistake is reading or writing coordinates in the wrong order.

Example 12.4.1
Identifying coordinates from descriptions
  • ProblemName the ordered pair for each location: (a) 5 units right, 2 units up; (b) 3 units left, 4 units down; (c) 6 units right, on the x-axis; (d) 7 units below the origin, on the y-axis.
  • ThinkMatch the direction to the sign of each coordinate. Right is positive x. Left is negative x. Up is positive y. Down is negative y. On the x-axis means y = 0. On the y-axis means x = 0.
  • Solve (a) (5, 2) (b) (−3, −4) (c) (6, 0) — on the x-axis, so y = 0 (d) (0, −7) — on the y-axis, so x = 0
  • CheckIn (a), both values are positive, matching "right" and "up." In (b), both are negative, matching "left" and "down." In (c), y is zero, confirming the x-axis. In (d), x is zero, confirming the y-axis.
Answer: (a) (5, 2), (b) (−3, −4), (c) (6, 0), (d) (0, −7)
Practice 12.4Ordered Pairs
  1. 4 units right, 3 units up
  2. 5 units left, 2 units up
  3. 3 units right, 6 units down
  4. 7 units left, 4 units down
  5. On the y-axis, 5 units above origin
  6. On the x-axis, 8 units left of origin
  7. At the origin
  8. 2 units right, on the x-axis
Section 12.5

Plotting Points

Plotting a point means marking its location on the coordinate plane. The process follows a simple, consistent sequence. Step 1: start at the origin (0, 0). Step 2: move horizontally by the x-coordinate, right if positive, left if negative, and not at all if zero. Step 3: from that position, move vertically by the y-coordinate, up if positive, down if negative, and not at all if zero. Step 4: mark the point with a dot and label it if required.

Think of the coordinate plane as a city grid. The x-axis is a street running east and west. The y-axis is an avenue running north and south. The origin is the intersection of Main Street and Central Avenue. An ordered pair gives directions: go three blocks east, then two blocks north. There is exactly one location that matches those directions, and marking it on the map is plotting the point.

Example 12.5.1
Plotting a set of points
  • ProblemPlot and label each point: A = (3, 4), B = (−2, 1), C = (0, −5), D = (−4, −3), E = (6, 0).
  • ThinkApply the four plotting steps to each point. Check the sign of each coordinate before moving.
  • Solve A (3, 4): 3 right, 4 up — upper right area B (−2, 1): 2 left, 1 up — upper left area C (0, −5): no move, 5 down — on the y-axis, below origin D (−4, −3): 4 left, 3 down — lower left area E (6, 0): 6 right, no move — on the x-axis, right of origin
  • CheckA is in the upper right. B is in the upper left. C is on the y-axis below origin. D is in the lower left. E is on the x-axis to the right.
Answer: All five points are placed at their correct coordinates. C and E lie on the axes.
Example 12.5.2
Reading coordinates from a graph
  • ProblemA coordinate plane has four labeled points. Find the ordered pair for each: P is 2 units right and 5 units above origin; Q is directly on the x-axis, 3 units left; R is 4 units below origin on the y-axis; S is 1 unit left and 7 units below origin.
  • ThinkRight is positive x. Left is negative x. Up is positive y. Down is negative y. On an axis means the other coordinate is zero.
  • Solve P: (2, 5) Q: (−3, 0) R: (0, −4) S: (−1, −7)
  • CheckEach pair matches the described direction and distance.
Answer: P is (2, 5), Q is (−3, 0), R is (0, −4), S is (−1, −7).
Practice 12.5Plotting Points

On a coordinate plane, plot and label each point.

  1. A = (3, 5)
  2. B = (−2, 4)
  3. C = (−4, −3)
  4. D = (5, −1)
  5. E = (0, 6)
  6. F = (−7, 0)
  7. G = (−1, −5)
  8. H = (4, 0)
Section 12.6

Quadrants

The x-axis and y-axis divide the coordinate plane into four regions. Each region is called a quadrant. The quadrants are numbered using Roman numerals, starting in the upper right and moving counterclockwise.

II
x: negative, y: positive
Upper left
I
x: positive, y: positive
Upper right
III
x: negative, y: negative
Lower left
IV
x: positive, y: negative
Lower right
Quadrantx-coordinatey-coordinateLocation
IPositivePositiveUpper right
IINegativePositiveUpper left
IIINegativeNegativeLower left
IVPositiveNegativeLower right
Points on the Axes

Points that lie on the x-axis or y-axis do not belong to any quadrant. They sit on the boundary between quadrants. The origin itself is the meeting point of all four quadrants but belongs to none of them.

Example 12.6.1
Identifying quadrants without plotting
  • ProblemIdentify the quadrant or axis location of each point: (a) (5, 3), (b) (−4, 7), (c) (−2, −9), (d) (8, −1), (e) (0, 6), (f) (−3, 0).
  • ThinkUse the signs of the coordinates. If either is zero, the point is on an axis.
  • Solve (a) (+, +) → Quadrant I (b) (−, +) → Quadrant II (c) (−, −) → Quadrant III (d) (+, −) → Quadrant IV (e) x = 0 → on the y-axis (f) y = 0 → on the x-axis
  • CheckEach result matches the sign pattern for that quadrant or axis.
Answer: (a) I, (b) II, (c) III, (d) IV, (e) y-axis, (f) x-axis
Practice 12.6Identifying Quadrants

State the quadrant or axis location of each point without plotting.

  1. (6, 4)
  2. (−3, 8)
  3. (−5, −1)
  4. (2, −9)
  5. (0, −4)
  6. (7, 0)
  7. (−6, −6)
  8. (1, 10)
Section 12.7

Coordinates and Direction

The sign of each coordinate carries directional meaning. The x-coordinate tells how far a point is from the y-axis and in which horizontal direction. A positive x-coordinate means the point is to the right of the y-axis. A negative x-coordinate means the point is to the left. An x-coordinate of zero means the point is on the y-axis itself.

The y-coordinate tells how far a point is from the x-axis and in which vertical direction. A positive y-coordinate means the point is above the x-axis. A negative y-coordinate means the point is below. A y-coordinate of zero means the point is on the x-axis.

The absolute value of each coordinate is the distance from the axis. The point (5, −3) is 5 units to the right of the y-axis and 3 units below the x-axis. This connects the absolute value concept from Chapter 1 directly to coordinate geometry: distance is always positive, even when the direction is negative.

Reflecting Points Across Axes

When the sign of one coordinate is changed, the point moves to the opposite side of that axis while staying the same distance from it. The point (3, 4) and the point (−3, 4) are reflections of each other across the y-axis. The point (3, 4) and the point (3, −4) are reflections across the x-axis. The point (3, 4) and (−3, −4) are reflections through the origin.

Example 12.7.1
Describing coordinates using direction language
  • ProblemDescribe the location of each point using direction words and distance from each axis: (a) (4, −6), (b) (−1, 3), (c) (−5, −2).
  • ThinkUse the x-coordinate for horizontal location relative to the y-axis. Use the y-coordinate for vertical location relative to the x-axis.
  • Solve (a) (4, −6): 4 right of y-axis, 6 below x-axis → Quadrant IV (b) (−1, 3): 1 left of y-axis, 3 above x-axis → Quadrant II (c) (−5, −2): 5 left of y-axis, 2 below x-axis → Quadrant III
  • CheckThe quadrant assignments match the sign patterns from Section 12.6.
Answer: (a) 4 right, 6 below, Quadrant IV. (b) 1 left, 3 above, Quadrant II. (c) 5 left, 2 below, Quadrant III.
Practice 12.7Coordinates and Direction
  1. For each point, state the distance from the y-axis, the distance from the x-axis, and the quadrant or axis location.

    (a) (−3, 7)    (b) (5, −2)    (c) (−4, −8)    (d) (0, 9)    (e) (6, 0)
  2. The point (3, 5) is reflected across the y-axis. What are the new coordinates?
  3. The point (−2, 4) is reflected across the x-axis. What are the new coordinates?
  4. The point (6, −1) is reflected through the origin. What are the new coordinates?
Sections 12.8 – 12.10
Tables, Patterns, and Pattern Rules
Tables of Values · Linear Patterns · Writing y = mx + b
Section 12.8

Tables of Values

Before graphing a pattern or relationship, it is helpful to organize the data in a table of values. A table of values is a structured list of input-output pairs. Each row of the table holds one ordered pair that can be plotted on the coordinate plane.

In a relationship between two quantities, the quantity that is freely chosen is called the input, usually represented by x and placed on the horizontal axis. The quantity that results from the input is called the output, usually represented by y and placed on the vertical axis.

A table of values typically has two columns. The left column holds input values, labeled x or with the name of the first quantity. The right column holds output values, labeled y or with the name of the second quantity. Each row is one pair of related values.

Example 12.8.1
Building a table of values from a rule
  • ProblemA worker earns $12 for each hour worked. Build a table of values for hours worked from 0 to 5.
  • ThinkThe rule is: earnings = 12 × hours. Let x represent hours and y represent earnings. For each x-value, multiply by 12 to find y.
  • Solve
    x (Hours)y (Earnings $)
    00
    112
    224
    336
    448
    560
  • CheckEach y-value is exactly 12 times its x-value. The increase from row to row is a constant $12.
Answer: Six ordered pairs: (0,0), (1,12), (2,24), (3,36), (4,48), (5,60).
Example 12.8.2
Building a table with negative values
  • ProblemA temperature reading changes according to the rule y = x − 3. Build a table of values for x from −2 to 3.
  • ThinkSubstitute each x-value into y = x − 3. Subtracting 3 from a negative number produces a more negative result.
  • Solve
    xy = x − 3
    −2−5
    −1−4
    0−3
    1−2
    2−1
    30
  • CheckEach y-value is exactly 3 less than the corresponding x-value throughout.
Answer: (−2,−5), (−1,−4), (0,−3), (1,−2), (2,−1), (3,0).
Student Tip — Always Include x = 0

Always include x = 0 in a table when possible. The y-value when x is zero is called the y-intercept, and it will become very important in Algebra I. For now, it is the point where a line or pattern crosses the y-axis.

Practice 12.8Tables of Values

Build a table of values for each rule using x-values from −2 to 4.

  1. y = x + 4
  2. y = 2x
  3. y = x − 3
  4. y = 4x + 1
  5. y = −x + 5
  6. y = 3x − 2
Section 12.9

Simple Linear Patterns

A pattern is a sequence of values that changes according to a consistent rule. A linear pattern is a pattern in which the output changes by the same amount each time the input increases by one. The values grow or shrink at a constant rate, and when graphed, a linear pattern produces a straight line.

To check whether a pattern is linear, look at the differences between consecutive output values. If those differences are all equal, the pattern is linear.

Example 12.9.1
Identifying a linear pattern
  • ProblemDetermine whether each table shows a linear pattern.

    Table A: x = 0,1,2,3,4 and y = 3,6,9,12,15.
    Table B: x = 0,1,2,3,4 and y = 1,2,4,8,16.
  • ThinkFind the differences between consecutive y-values in each table.
  • Solve Table A differences: 6−3=3, 9−6=3, 12−9=3, 15−12=3 All differences equal 3. → Linear pattern. Table B differences: 2−1=1, 4−2=2, 8−4=4, 16−8=8 Differences are 1, 2, 4, 8. Not constant. → Not linear.
  • CheckTable A has constant differences. Table B has differences that double each time, indicating exponential growth.
Answer: Table A is linear. Table B is not linear.
Example 12.9.2
A decreasing linear pattern
  • ProblemA candle is 20 cm tall when first lit and burns down 2 cm each hour. Complete the table and identify the pattern.
  • ThinkThe candle loses 2 centimeters per hour. Subtract 2 from each previous height.
  • Solve
    x (Hours)y (Height cm)
    020
    118
    216
    314
    412
    Differences: 18−20=−2, 16−18=−2, 14−16=−2, 12−14=−2 Constant difference of −2.
  • CheckAll differences are −2, confirming a linear decrease.
Answer: The pattern is linear with a constant decrease of 2 centimeters per hour.
Key Idea — Why Constant Differences Mean a Straight Line

When a pattern adds the same amount to the output each time the input increases by one, the graph produces a straight line. The constant difference is related to the steepness of that line, a concept Algebra I formalizes as slope. A negative constant difference means the line slopes downward from left to right.

Practice 12.9Identifying Linear Patterns

Find the constant difference for each set of y-values. State whether the pattern is linear.

  1. {(0,2), (1,5), (2,8), (3,11), (4,14)}
  2. {(0,1), (1,3), (2,9), (3,27), (4,81)}
  3. {(0,10), (1,7), (2,4), (3,1), (4,−2)}
  4. {(0,0), (1,4), (2,7), (3,11), (4,16)}
  5. A phone battery is at 100% and drops 15% per hour. List y-values for hours 0 through 4. Are the differences constant?
Section 12.10

Pattern Rules

A pattern rule is a statement or formula that describes how to find any output value from a given input value. To write a pattern rule from a table of values, ask two questions. First: by how much does y change each time x increases by 1? This is the constant difference, sometimes called the rate of change. Second: what is the y-value when x is 0? This is the starting value.

If the y-value starts at b when x is 0, and increases by m for each unit increase in x, the rule is:

Pattern Rule (Linear)

y = mx + b

where m = the constant rate of change (how much y changes per unit increase in x), and b = the starting value (the value of y when x = 0). This formula will be recognized in Algebra I as the slope-intercept form of a linear equation.

Example 12.10.1
Writing a pattern rule from a table
  • ProblemWrite the pattern rule for the table: x = 0,1,2,3,4 and y = 5,8,11,14,17.
  • ThinkFind the constant difference and identify the y-value when x is 0.
  • Solve Constant difference: 8 − 5 = 3. Rate of change m = 3. Starting value: when x = 0, y = 5. So b = 5. Pattern rule: y = 3x + 5
  • CheckAt x = 2: y = 3(2) + 5 = 11. Table shows 11. ✓ At x = 4: y = 3(4) + 5 = 17. Table shows 17. ✓
Answer: y = 3x + 5
Example 12.10.2
Writing a rule for a decreasing pattern
  • ProblemWrite the pattern rule for the table: x = 0,1,2,3,4 and y = 20,17,14,11,8.
  • ThinkFind the constant difference and starting value.
  • Solve Constant difference: 17 − 20 = −3. Rate of change m = −3. Starting value: when x = 0, y = 20. So b = 20. Pattern rule: y = −3x + 20
  • CheckAt x = 3: y = −3(3) + 20 = −9 + 20 = 11. Table shows 11. ✓
Answer: y = −3x + 20
Example 12.10.3
Using a rule to find unknown values
  • ProblemUsing the rule y = 3x + 5 from Example 12.10.1, find the output when x = 10 and when x = 50.
  • Solve x = 10: y = 3(10) + 5 = 30 + 5 = 35 x = 50: y = 3(50) + 5 = 150 + 5 = 155
  • CheckFrom x = 4 (y = 17) to x = 10 is 6 more steps: 17 + 6(3) = 17 + 18 = 35. Confirmed.
Answer: When x = 10, y = 35. When x = 50, y = 155.
Practice 12.10Writing Pattern Rules

Write the pattern rule in the form y = mx + b for each table.

Problem 1
xy
06
19
212
315
Problem 2
xy
00
15
210
315
Problem 3
xy
010
18
26
34
Problem 4
xy
0−1
12
25
38
  1. Using the rule y = 4x − 3, build a table for x = 0 to x = 5. Then find the output when x = 20.
  2. A pattern has a starting value of 2 and increases by 7 each time x increases by 1. Write the rule. Find the output when x = 8.
Sections 12.11 – 12.14
Graphing, Reading, Mistakes, and the Road Ahead
Graphing Patterns · Reading Graphs · Common Mistakes · Preparing for Algebra I
Section 12.11

Graphing Patterns

Once a table of values is complete, the next step is to plot the ordered pairs on a coordinate plane and observe the shape they form. For a linear pattern, the plotted points will fall along a straight line. Connecting them confirms the pattern and makes the relationship visible. The steps are: build or review the table of values, set up a coordinate plane with an appropriate scale, plot each ordered pair, observe whether the points fall in a line, and if the pattern is linear, connect the points with a straight line and extend it beyond the plotted points.

What the Graph Reveals

A line rising from left to right shows a positive rate of change. A line falling from left to right shows a negative rate of change. A steep line means the output changes quickly. A shallow line means the change is slow. The point where the line crosses the y-axis (when x is 0) is the starting value of the pattern.

Example 12.11.1
Graphing a linear pattern
  • ProblemGraph the pattern y = 2x + 3 using the ordered pairs from the table.
  • ThinkThe constant difference is +2 per unit increase in x. The starting value (b) is 3. All points should fall on a straight line.
  • Solve
    xy = 2x + 3
    −2−1
    −11
    03
    15
    27
    39
    Plot: (−2,−1), (−1,1), (0,3), (1,5), (2,7), (3,9). All six points align in a straight line rising from lower left to upper right. The line crosses the y-axis at y = 3 and the x-axis between x = −1 and x = −2.
  • CheckAt x = −2: y = 2(−2) + 3 = −4 + 3 = −1. ✓ At x = 3: y = 2(3) + 3 = 9. ✓
Answer: A straight line rising from lower left to upper right, crossing the y-axis at 3. Rate of change is 2.
Example 12.11.2
Graphing a decreasing pattern — phone battery
  • ProblemA phone battery starts at 100% and loses 15% per hour. Graph the battery level from hour 0 to hour 6.
  • ThinkThe rule is y = −15x + 100, where x is hours and y is battery percentage.
  • Solve
    x (Hours)y (Battery %)
    0100
    185
    270
    355
    440
    525
    610
    Points fall on a straight line running downward from (0, 100) to (6, 10). Scale x-axis: 0 to 7. Scale y-axis: 0 to 110 by 20s.
  • CheckAt x = 3: y = −15(3) + 100 = −45 + 100 = 55. Table shows 55. ✓
Answer: A straight downward line from (0, 100) to (6, 10). The battery decreases at 15% per hour. The line reaches zero at about hour 6.67.
Student Note — Graphs Reveal What Tables Hide

The graph of the battery example also shows when the battery would reach zero, even without computing it. By following where the line would cross the x-axis, a student can see approximately hour 6.67. This is one of the most important reasons graphs are useful: they reveal information that lies between the values in a table.

Practice 12.11Graphing Practice
  1. Build a table of values for y = 2x − 1 using x-values from −2 to 4. Plot all points. Connect with a straight line. Describe the direction and starting point of the graph.
  2. Build a table for y = −x + 4 using x-values from −1 to 5. Where does the line cross the y-axis? Where does it cross the x-axis?
  3. A car travels at 55 mph. Build a table showing distance after 0 through 5 hours. Write the rule. Graph the relationship. Identify the starting value and rate of change.
  4. A plant is 4 inches tall when first measured and grows 1.5 inches per week. Build a table for weeks 0 through 6. Write the rule. Graph the pattern. Use the graph to estimate when the plant will reach 12 inches.
Section 12.12

Reading Meaning from Graphs

Plotting a graph is only the first step. Equally important is reading a graph, interpreting what it shows, and drawing conclusions from its shape. The y-intercept is the point where the graph crosses the y-axis. It shows the output when the input is zero and often represents a starting value in context: beginning temperature, initial cost, or battery level at hour zero. The x-intercept is the point where the graph crosses the x-axis. It shows the input when the output is zero and may represent the moment something runs out, reaches zero, or breaks even.

The direction of the line shows whether the relationship is increasing or decreasing. The steepness shows how fast. Equal spacing between points confirms a linear relationship. Points that drift away from a line suggest a non-linear one.

Example 12.12.1
Reading information from a graph
  • ProblemA line passes through (0, 20) and (5, 45). Answer: (a) starting value, (b) rate of change, (c) output when x = 3, (d) when does output reach 50?
  • ThinkIdentify the y-intercept and compute the rate of change from the two given points. Then write the rule and use it.
  • Solve (a) y-intercept: (0, 20). Starting value = 20. (b) Change in y: 45 − 20 = 25. Change in x: 5 − 0 = 5. Rate of change: 25 ÷ 5 = 5 per unit. Rule: y = 5x + 20. (c) x = 3: y = 5(3) + 20 = 15 + 20 = 35. (d) Set y = 50: 50 = 5x + 20 → 30 = 5x → x = 6.
  • CheckAt x = 5: y = 5(5) + 20 = 45. Matches the given point. ✓
Answer: (a) 20, (b) 5 per unit, (c) 35, (d) when x = 6.
Example 12.12.2
Comparing two patterns on one graph
  • ProblemStudent A starts with $10 and saves $20 per week. Student B starts with $70 and saves $5 per week. At what week are their savings equal?
  • ThinkWrite rules for each student, build combined tables, and find where the values match.
  • Solve Student A: y = 20x + 10 Student B: y = 5x + 70 Week 0: A = $10, B = $70 Week 1: A = $30, B = $75 Week 2: A = $50, B = $80 Week 3: A = $70, B = $85 Week 4: A = $90, B = $90 ← Equal at week 4
  • CheckA: 20(4) + 10 = 90. B: 5(4) + 70 = 90. Both equal $90 at week 4. ✓
Answer: Their savings are equal at week 4 ($90 each). On the graph, the two lines intersect at (4, 90).
Practice 12.12Reading Meaning from Graphs
  1. A graph shows a line passing through (0, 30) and (6, 0). What is the starting value? What is the rate of change? Write the rule. What does the x-intercept mean in context?
  2. A taxi charges a flat fee of $3 plus $2.50 per mile. Write the rule for total cost in terms of miles. Build a table for 0 to 6 miles. What is the cost for a 10-mile ride? What does the y-intercept represent?
  3. A tank holds 150 gallons and drains at 25 gallons per minute. Write the rule for gallons remaining. Build a table from 0 to 6 minutes. When will the tank be empty? How does the graph show this?
  4. Two phone plans: Plan A costs $30 per month plus $0.10 per text. Plan B costs $50 per month with unlimited texts. Build tables for 0 to 300 texts in increments of 50. At what number of texts do both plans cost the same? Graph both on one coordinate plane.
Section 12.13

Common Coordinate Mistakes

Mistake 1 — Reversing the Coordinates
The point 3 right, 5 up written as (5, 3)

Correction: The x-coordinate always comes first. "How far horizontally?" goes first. A point 3 right and 5 up is (3, 5), not (5, 3).

Mistake 2 — Misreading the Sign of a Coordinate
Plotting (−3, 4) at (3, 4) by forgetting to move left

Correction: Before plotting any point, check the sign of each coordinate. Negative x means move left. Negative y means move down.

Mistake 3 — Inconsistent Scale on an Axis
Y-axis marks at 0, 5, 10, 20, 30 at equal spacing

Correction: Use equal spacing for all marks on each axis. An inconsistent scale distorts a straight-line relationship and makes a linear pattern appear curved.

Mistake 4 — Mixing Up Table Columns
Placing x-values in the right column and y-values in the left column

Correction: Always label columns clearly. The input (x) belongs in the left column. The output (y) belongs in the right column.

Mistake 5 — Skipping the Check Step
Proceeding to graph without verifying table values against the rule

Correction: After building the table, verify at least two rows by substituting back into the rule. One arithmetic error in the table will produce a point that does not align with the others.

Mistake 6 — Connecting Non-Linear Points with a Straight Line
Drawing a straight line through points that are not collinear

Correction: Check constant differences in the table before connecting points. If the differences are not constant, the pattern is not linear and the points should not be connected with a straight line.

Mistake 7 — Confusing Which Axis Is Which
Treating the vertical axis as the x-axis because the graph is not labeled

Correction: The horizontal axis is always the x-axis and runs left-right. The vertical axis is always the y-axis and runs up-down. Confirm labels before plotting.

Practice 12.13Error Analysis

Each problem below contains an error. Identify it, explain what went wrong, and write the correct answer.

  1. A student is asked to plot (3, −5). The student starts at the origin, moves 5 units down, then 3 units to the right. Is the point in the right location? Explain the error and describe the correct procedure.
  2. A student writes that the point (−4, 7) is in Quadrant IV. Identify the error and state the correct quadrant.
  3. A student builds the following table for y = 3x + 2: at x = 0, y = 2; at x = 1, y = 5; at x = 2, y = 9; at x = 3, y = 12; at x = 4, y = 15. One value is incorrect. Which row? What likely went wrong? Give the correct value.
  4. A student writes the rule for the table x = 0,1,2,3 and y = 4,7,10,13 as y = 4x + 3. Show why this is incorrect and write the correct rule.
  5. A student graphs y = 2x + 1 and plots (1, 3), (2, 5), (3, 8), (4, 9). Two points are wrong. Identify which ones, give the correct y-values, and explain the likely errors.
  6. A student says the ordered pair (0, 5) is the same as the origin because one coordinate is zero. Explain why this is incorrect and describe where (0, 5) is actually located.
Section 12.14

Preparing for Linear Equations

Chapter 12 has introduced the tools and language that Algebra I will use constantly. Students in this chapter have learned to locate any point on the coordinate plane using a signed ordered pair, read coordinates from a graph and plot them on a grid, build tables of values by applying a rule, recognize linear patterns by identifying constant differences, write a pattern rule in the form y = mx + b, and graph a linear pattern and read information from it.

All of these skills are pre-algebraic foundations for the study of linear equations.

What Algebra I Will Add

In Algebra I, students will formalize what has been explored here. The pattern rule y = mx + b will be named the slope-intercept form of a linear equation. The constant difference m will be named the slope. The starting value b will be named the y-intercept. Students will solve for equations given two points, graph equations without a table, and work with systems of two equations simultaneously. Every one of these algebra skills begins with the coordinate plane and ordered pairs.

A Preview of Slope

In Algebra I, the constant difference in a table becomes formalized as slope:

slope = (change in y) ÷ (change in x)

A positive slope means the line rises. A negative slope means the line falls. A slope of zero means the line is horizontal. Students will compute slope from two points, from a table, and from an equation. The constant differences practiced in this chapter are already the concept of slope — just not yet named.

Practice 12.14Readiness Check
  1. Build a table for y = 4x − 3 using x = −1 to 4. Graph the points. Describe the direction of the line and where it crosses the y-axis.
  2. A pattern rule is y = −2x + 10. Build a table for x = 0 to x = 5. At what value of x does y equal zero? What does this represent on the graph?
  3. A line on a coordinate plane passes through (0, 8) and (4, 0). What is the starting value? What is the rate of change? Write the pattern rule.
  4. Two students are saving money. Student A starts with $10 and saves $20 per week. Student B starts with $70 and saves $5 per week. Write rules for each. Build a combined table for weeks 0 through 8. At what week do their savings become equal?
  5. A student says that any pattern that starts at zero must have a rule of the form y = mx. Is this always true? Give an example and explain.
Chapter 12 — Back Matter

Chapter Summary

Key Terms Review

coordinate plane
A flat surface defined by two perpendicular number lines. Every point is located by two coordinates.
x-axis
The horizontal number line on the coordinate plane. Positive values are to the right of the origin. Negative values are to the left.
y-axis
The vertical number line on the coordinate plane. Positive values are above the origin. Negative values are below.
origin
The point where the x-axis and y-axis intersect. Its coordinates are (0, 0). Every other point is measured from here.
ordered pair
Two numbers in the form (x, y) that name the location of a point. The x-coordinate always comes first.
x-coordinate
The first number in an ordered pair. Describes the horizontal position of the point relative to the y-axis.
y-coordinate
The second number in an ordered pair. Describes the vertical position of the point relative to the x-axis.
quadrant
One of the four regions of the coordinate plane. Numbered I through IV counterclockwise from the upper right.
plot
To mark a point on the coordinate plane at the location described by an ordered pair.
table of values
An organized list of input-output pairs used to build coordinates for graphing. Inputs in the left column, outputs in the right.
pattern
A sequence of values that changes according to a consistent rule.
linear pattern
A pattern in which the output changes by the same constant amount each time the input increases by one. Produces a straight line when graphed.
pattern rule
A formula that describes how to calculate any output from a given input. Written as y = mx + b for linear patterns.
input
The independent value in a relationship. Usually represented by x and placed on the horizontal axis.
output
The dependent value in a relationship. Usually represented by y and placed on the vertical axis.
graph
A visual representation of a set of ordered pairs or a mathematical relationship on the coordinate plane.

Concept Review Questions

Answer in complete sentences where appropriate.

  1. Why is the coordinate plane more useful than a single number line for representing certain kinds of information?
  2. What does the origin represent on the coordinate plane, and what are its coordinates?
  3. A classmate says that the ordered pairs (4, 7) and (7, 4) name the same point. Is this correct? Explain.
  4. In which quadrant would you find a point whose x-coordinate is negative and whose y-coordinate is positive? Describe in words where that quadrant is located on the plane.
  5. A point lies on the x-axis. What do you know about one of its coordinates? Which one, and what is its value?
  6. A point lies on the y-axis. What do you know about one of its coordinates?
  7. Explain the difference between the x-coordinate and the y-coordinate in terms of direction of movement from the origin.
  8. What does it mean for a pattern to be linear? Give an example of a pattern that is linear and one that is not linear, using tables of values.
  9. What is a pattern rule, and what two pieces of information do you need to write a pattern rule for a linear pattern?
  10. A student builds a table of values and plots the points on a coordinate plane. The points do not fall along a straight line. What are two possible explanations?
  11. What does the y-intercept of a graph represent in a real-world context? Use the candle-burning example to explain.
  12. Why is it important to use a consistent scale on both axes when graphing?

Computation and Skill Practice

Part AOrdered Pairs and Quadrants

Write the ordered pair for each described location.

  1. 4 right, 3 up
  2. 5 left, 2 up
  3. 3 right, 6 down
  4. 7 left, 4 down
  5. y-axis, 5 above origin
  6. x-axis, 8 left of origin
  7. At the origin
  8. 2 right, on x-axis

State the quadrant or axis location of each point.

  1. (6, 4)
  2. (−3, 8)
  3. (−5, −1)
  4. (2, −9)
  5. (0, −4)
  6. (7, 0)
  7. (−6, −6)
  8. (1, 10)
Part BTables of Values

Build a table for each rule using x-values from −2 to 4.

  1. y = x + 4
  2. y = 2x
  3. y = x − 3
  4. y = 4x + 1
  5. y = −x + 5
  6. y = 3x − 2
Part CPattern Rules

Find the constant difference. State whether the pattern is linear. If linear, write the rule.

  1. y-values: 3, 7, 11, 15, 19 (for x = 0,1,2,3,4)
  2. y-values: 20, 17, 14, 11, 8 (for x = 0,1,2,3,4)
  3. y-values: 1, 2, 4, 8, 16 (for x = 0,1,2,3,4)
  4. y-values: 0, 4, 8, 12, 16 (for x = 0,1,2,3,4)
  5. y-values: 6, 6, 6, 6, 6 (for x = 0,1,2,3,4)

Word Problems

For each problem: define your variable, write a pattern rule, build a table, answer the specific question, and describe what the graph would look like.

  1. A gym membership costs $50 per month. Write an expression for total cost after any number of months. How much after one full year? What is the y-intercept, and what does it mean?
  2. A car travels at a constant speed of 55 miles per hour. Write an expression for distance in t hours. How far in 3.5 hours? Where does the graph cross the y-axis?
  3. A water tank holds 150 gallons and drains at 25 gallons per minute. Write the rule for gallons remaining. Build a table from 0 to 6 minutes. When will the tank be empty? How can the graph show this?
  4. Two students save money. Student A starts with $20 and saves $10 per week. Student B starts with $0 and saves $15 per week. Build a table for each student for weeks 0 through 8. Plot both on the same coordinate plane. Find the week where both students have the same amount saved.
  5. A plumber charges $75 per hour plus a $50 service call fee. Write a rule for total cost. Build a table for 0 to 5 hours. For how many hours of work does the total cost equal $275?
  6. A candle is 18 inches tall when first lit and burns at 2 inches per hour. Write the rule. Build a table for hours 0 through 9. At what hour does the candle burn out? Describe the shape of the graph.

Error Analysis

Each problem shows a student's work. Identify the error and write the correct answer.

Problem 1
A student plots (3, −5) by moving 5 units down, then 3 right from the origin.

What is wrong with the sequence? Describe the correct procedure step by step.

Problem 2
A student says (−4, 7) is in Quadrant IV.

Identify the error. State the correct quadrant and explain the sign pattern that determines it.

Problem 3
For y = 3x + 2, a student builds the table: x = 0,1,2,3,4 and y = 2,5,9,12,15.

Which value is wrong? What should it be? What likely arithmetic error occurred?

Problem 4
For the table x = 0,1,2,3 and y = 4,7,10,13, a student writes the rule y = 4x + 3.

Show why this is incorrect by testing with x = 1. Write the correct rule.

Problem 5
A student says (0, 5) is the same as the origin because one coordinate is zero.

Explain why this is wrong. Where is (0, 5) actually located on the coordinate plane?

Chapter 12 Checkpoint

Chapter 12 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 12
Part 1 — Coordinate Plane Knowledge
  1. Name the four quadrants and give the sign pattern (positive or negative) for each coordinate in each quadrant.
  2. What are the coordinates of the origin?
  3. A point has a negative x-coordinate and a positive y-coordinate. In which quadrant does it lie?
  4. A point is located 4 units to the right and 7 units below the origin. Write its ordered pair and name its quadrant.
  5. A point lies on the y-axis, 3 units above the origin. Write its ordered pair.
Part 2 — Plotting and Reading
  1. Plot and label the following points on a coordinate plane: (2, 4), (−3, 1), (0, −5), (−4, −2), (5, 0).
  2. A coordinate plane has a point labeled W at 3 units to the left and 6 units up from the origin. What is the ordered pair for W?
  3. A point is reflected across the y-axis from (5, −3). What are the new coordinates?
Part 3 — Tables of Values
  1. Build a table of values for y = 4x − 3 using x-values from −1 to 4.
  2. Determine whether the following y-values form a linear pattern: 0, 1, 4, 9, 16 (for x = 0,1,2,3,4). Explain your answer using differences.
  3. Determine whether the following table represents a linear pattern. If so, write the rule: x = 0,1,2,3,4 and y = 7,11,15,19,23.
Part 4 — Pattern Rules and Graphing
  1. Write the pattern rule for a pattern with a starting value of 6 that increases by 4 each time x increases by 1.
  2. A pattern rule is y = −2x + 10. Build a table for x = 0 to x = 5. At what value of x does y equal zero?
  3. Graph the pattern y = 3x − 1 for x-values from −1 to 4. Describe the direction and starting point of the line.
Part 5 — Reading Meaning from Graphs and Applications
  1. A line passes through (0, 8) and (4, 0). What is the starting value? What is the rate of change? Write the pattern rule.
  2. A candle is 18 inches tall when first lit and burns at 2 inches per hour. Write the rule, build a table for 0 to 9 hours, and determine when the candle burns out completely.
  3. Two students are saving money. Student A starts with $10 and saves $20 per week. Student B starts with $70 and saves $5 per week. Write rules for each. Build a combined table for weeks 0 through 8. At what week are their savings equal?
Part 6 — Error Analysis
  1. A student says the ordered pair for a point 5 units left and 3 units down is (3, −5). Identify and correct the error.
  2. A student plots (−2, 4) by moving 2 units up and then 4 units to the left. Describe the correct procedure and explain what is wrong with the student's method.

Answers may be placed in the answer key.

Looking Ahead
Chapter 13: Word Problems and Problem Solving

Chapter 12 gave you the visual language of mathematics. Chapter 13 will bring together arithmetic, fractions, decimals, percents, variables, equations, and the coordinate plane in realistic problem-solving situations. The focus will shift from learning individual skills to applying them together. You will practice reading mathematical information from words, choosing the right operations, drawing diagrams, defining variables, writing equations, and explaining answers in context. Every skill developed in Chapters 1 through 12 will be called on. Chapter 13 is where all of those tools become a unified problem-solving method.

MGU Library Connection
  • Algebra I, Chapter 1: Formal definitions of slope, slope-intercept form, and standard form of a linear equation
  • Algebra I, Chapter 5: Functions and function notation — pattern rules become function rules
  • MGU Geometry: Coordinate geometry, distance formula, midpoint formula, parallel and perpendicular lines
  • Appendix J: Algebra I Readiness Check — assess coordinate plane preparation before beginning Algebra I
  • MGU Reference Tables: Blank coordinate plane grid for practice; formula sheet including y = mx + b
  • MGU Dictionary entries: coordinate plane, ordered pair, slope, y-intercept, linear equation