MGU Mathematics Series — Algebra I
Part III: Functions, Graphs, and Linear Models
Coordinate Plane · Intercepts · Slope · Graphing Relationships
Chapter 6 of 14
MGU Mathematics Series  ·  Algebra I  ·  Library Textbook Edition

Chapter 6:
The Coordinate Plane
and Graphing Relationships

Ordered pairs, quadrants, intercepts, slope as rate of change, and graphs from tables and verbal descriptions — the visual language of algebraic relationships

■ Math Level: Algebra I ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapter 5: Introduction to Functions  ·  Prepares students for Chapter 7: Linear Functions and Equations

Chapter 6 — Front Matter

Purpose and Overview

This chapter deepens your graphing skills and helps you understand graphs as visual representations of relationships between quantities. You will review the coordinate plane, ordered pairs, axes, and quadrants, and you will extend that foundation by studying intercepts, slope, and how graphs communicate meaning. By the end of this chapter, you should be able to read a graph, build a graph from a table or description, calculate slope as a rate of change, and interpret what slope and intercepts mean in real situations.

Graphs are not decoration. A graph shows how two quantities relate to each other, how one changes as the other changes, where a relationship starts, and how fast it grows or falls. Learning to read and build graphs is one of the most important skills in algebra, and it will serve you in every mathematics and science course that follows.

Opening Question

A cell phone plan charges a flat monthly fee of $20 plus $0.05 per text message. If you send 100 texts in a month, your bill is $25. If you send 300 texts, your bill is $35. If you send 0 texts, your bill is exactly $20. How could you show this relationship visually? If you wanted to predict the bill for 500 texts without calculating, where would you look? How steep would the pattern be if the cost per text went up to $0.10? These are graphing questions. The answers live not only in equations, but in pictures of relationships.

14Sections
13Objectives
18Key Terms
12+Worked Examples

Learning Objectives

  1. 1Identify the parts of the coordinate plane, including axes, origin, and quadrants.
  2. 2Write and interpret ordered pairs.
  3. 3Plot points accurately on the coordinate plane.
  4. 4Read and interpret graphs in context.
  5. 5Identify x-intercepts and y-intercepts from a graph and from an equation.
  6. 6Calculate the change in x and change in y between two points.
  7. 7Calculate slope as a ratio of vertical change to horizontal change.
  8. 8Classify slope as positive, negative, zero, or undefined.
  9. 9Build a graph from a table of values.
  10. 10Build a graph from a verbal description.
  11. 11Graph real-world data and describe its pattern.
  12. 12Recognize and correct common coordinate and slope mistakes.
  13. 13Prepare for the formal study of linear functions and equations.
Chapter 6 — Contents

Sections at a Glance

Sections 6.1 – 6.5
The Coordinate Plane: Structure and Reading
Why Graphs Matter · Coordinate Plane · Ordered Pairs · Plotting · Reading Graphs
Section 6.1

Why Graphs Matter

Arithmetic is the study of numbers and operations on numbers. Algebra extends arithmetic by asking a deeper question: what if we do not know the number, or what if the number can change? Before the coordinate plane existed as a formal tool, mathematicians, scientists, and merchants tracked quantities in tables and words. A table could tell you that a plant was 3 centimeters tall after one week, 7 centimeters tall after two weeks, and 12 centimeters tall after three weeks. But a table requires work to see the pattern. You have to read the numbers carefully, compare them mentally, and build a picture in your mind.

A graph does that work automatically. The moment you plot those three data points and connect them with a line or curve, the pattern jumps off the page. You can see instantly whether growth is steady or accelerating. You can see where the plant started. You can predict future heights by continuing the pattern visually.

Graphs have a second advantage beyond convenience: they reveal structure that equations and tables sometimes hide. A straight line on a graph tells you the relationship is linear, meaning the change is constant. A curve tells you the change is not constant, which signals a different kind of relationship. Two lines crossing on a graph show the moment when two quantities become equal, which is a solution to a system of equations.

The coordinate plane is the tool that makes this possible. The inventor most credited with formalizing this system was René Descartes, a seventeenth-century French mathematician. His name lives on in the term Cartesian plane, another name for the coordinate plane. Descartes recognized that combining algebra with geometry gave mathematicians tools neither subject had alone.

MGU Library Connection
  • MGU Mathematics Dictionary entry on Coordinate Geometry
  • MGU Mathematics History supplement: Descartes and the Cartesian Plane
Section 6.2

The Coordinate Plane

The coordinate plane is built from two number lines. The x-axis runs horizontally, like the horizon. The y-axis runs vertically, like a wall. These two lines intersect at a point called the origin, written \((0, 0)\). Numbers to the right of the origin on the x-axis are positive; numbers to the left are negative. Numbers above the origin on the y-axis are positive; numbers below are negative.

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

QuadrantLocationx-signy-sign
IUpper rightPositivePositive
IIUpper leftNegativePositive
IIILower leftNegativeNegative
IVLower rightPositiveNegative
Diagram Instruction

Draw a coordinate plane. Draw the x-axis as a horizontal arrow extending left and right, labeled with values from −5 to 5. Draw the y-axis as a vertical arrow extending up and down, labeled with values from −5 to 5. Label the origin at (0, 0). Label each quadrant: Quadrant I in the upper right, Quadrant II in the upper left, Quadrant III in the lower left, Quadrant IV in the lower right. Use light grid lines to show the square grid.

Memory Device

Start in Quadrant I and move counterclockwise. In Quadrant I both signs are positive (+, +). In Quadrant II only x becomes negative (−, +). In Quadrant III both are negative (−, −). In Quadrant IV only y becomes negative (+, −).

Section 6.3

Ordered Pairs and Quadrants

An ordered pair is a pair of numbers written in the form \((x, y)\). The first number is the x-coordinate; the second is the y-coordinate. The order matters: \((3, 5)\) and \((5, 3)\) are different points. A positive x-coordinate means move right; negative means move left. A positive y-coordinate means move up; negative means move down. The quadrant is determined by the signs of both coordinates.

Example 6.3.1
Identifying quadrants from ordered pairs
  • ProblemIdentify the quadrant or axis location of each point: (a) \((4, 7)\)  (b) \((-3, 5)\)  (c) \((-2, -6)\)  (d) \((5, -1)\)  (e) \((0, 3)\)  (f) \((-4, 0)\)
  • ThinkLook at the signs of both coordinates. If a coordinate is zero, the point is on an axis.
  • Solve (a) (4, 7): Both positive → Quadrant I (b) (-3, 5): x negative, y positive → Quadrant II (c) (-2, -6): Both negative → Quadrant III (d) (5, -1): x positive, y negative → Quadrant IV (e) (0, 3): x is zero → On the y-axis (f) (-4, 0): y is zero → On the x-axis
  • CheckEach answer follows directly from the sign pattern. Points on an axis have at least one coordinate equal to zero. ✓
Section 6.4

Plotting Points

Plotting a point means finding and marking its exact location on the coordinate plane. The process has a fixed sequence: start at the origin, move horizontally by the x-coordinate, then move vertically by the y-coordinate.

How to Plot a Point

Step 1 — Start at the origin, (0, 0).

Step 2 — Move right if x is positive; move left if x is negative. Move the number of units given by the x-coordinate.

Step 3 — From that position, move up if y is positive; move down if y is negative.

Step 4 — Mark the point and label it with its ordered pair if needed.

Example 6.4.1
Plotting points on a coordinate plane
  • ProblemPlot and label: \(A = (3, 4)\), \(B = (-2, 5)\), \(C = (-4, -3)\), \(D = (6, -1)\), \(E = (0, -2)\).
  • ThinkFollow the plotting procedure for each point, noting sign and direction.
  • Solve A: right 3, up 4 → Quadrant I B: left 2, up 5 → Quadrant II C: left 4, down 3 → Quadrant III D: right 6, down 1 → Quadrant IV E: no horizontal move, down 2 → on the y-axis
  • CheckVerify each point is in the expected quadrant and matches the sign pattern. ✓
Answer: Points plotted as described. E lies on the y-axis, not in any quadrant.
Diagram Instruction

Draw a coordinate plane with axes from −6 to 6. Plot and label all five points. Show the horizontal and vertical movement from the origin as light dashed lines for each point.

Student Warning

Always move horizontally first, then vertically. Students who vary this process often make sign errors. Building a consistent habit now prevents confusion in later chapters.

Section 6.5

Reading Graphs in Context

Plotting isolated points is a starting skill. More important is reading what a graph communicates about a real relationship. When you read a graph in context, ask: What are the axes? What quantity does each axis represent, and what units? What is the shape — straight line or curve? What do specific points mean?

Example 6.5.1
Interpreting a graph in context
  • ProblemA graph shows candle height (cm) over time (hours). The graph is a straight line from \((0, 20)\) to \((10, 0)\). Interpret (a) the point \((0, 20)\), (b) the point \((10, 0)\), and (c) the overall shape.
  • Thinkx-axis = time in hours. y-axis = height in centimeters. Read each ordered pair as a statement about the candle.
  • Solve (a) (0, 20): At 0 hours, the candle is 20 cm tall. Starting height. (b) (10, 0): At 10 hours, the candle is 0 cm. Fully burned. (c) Straight line falling → constant burn rate, linear decrease.
  • CheckBoth points match the given information. The straight line confirms constant rate of burning. ✓
Answer: The candle starts at 20 cm and burns at a constant rate, reaching zero at 10 hours.
Tip

Every graph problem has a context. Never ignore the axis labels. Numbers alone mean nothing without knowing what quantity they represent.

Practice 6.1 – 6.5Coordinate Plane and Reading Graphs
  1. Identify the quadrant or axis location: (a) \((-5, 2)\)  (b) \((3, -7)\)  (c) \((0, -4)\)  (d) \((-1, -1)\)  (e) \((6, 0)\)
  2. Plot and label the following points on a single coordinate plane: \(P = (4, 2)\), \(Q = (-3, 5)\), \(R = (-5, -1)\), \(S = (2, -4)\), \(T = (0, 3)\), \(U = (-2, 0)\).
  3. A graph shows savings account balance (dollars) on the y-axis and months on the x-axis. The graph passes through \((0, 500)\), \((3, 650)\), and \((6, 800)\). (a) What does the point \((0, 500)\) mean? (b) What does the slope direction tell you about the account?
  4. A graph is a straight line from \((0, 30)\) to \((6, 0)\). If the x-axis is days and the y-axis is gallons of fuel in a tank, interpret both intercepts in context.
Sections 6.6 – 6.7
Intercepts and Measuring Change
x-Intercepts · y-Intercepts · Finding from Graphs · Finding from Equations · Δx and Δy
Section 6.6

x-Intercepts and y-Intercepts

Two special points on any graph are the intercepts — the points where the graph crosses the axes. The y-intercept is where the graph crosses the y-axis. On the y-axis, the x-coordinate is always zero. Therefore, the y-intercept occurs where \(x = 0\), and it is written as \((0, b)\). In many applications, the y-intercept represents the starting value of a quantity before any change occurs.

The x-intercept is where the graph crosses the x-axis. On the x-axis, the y-coordinate is always zero. Therefore, the x-intercept occurs where \(y = 0\), and it is written as \((a, 0)\). In applications, the x-intercept often represents the moment a quantity reaches zero — when a balance is paid off, when a candle burns out, or when a resource runs out.

Finding Intercepts from an Equation

y-intercept — Set \(x = 0\) in the equation and solve for y. The y-intercept is \((0, y)\).

x-intercept — Set \(y = 0\) in the equation and solve for x. The x-intercept is \((x, 0)\).

Example 6.6.1
Identifying intercepts from a graph
  • ProblemA straight line passes through \((0, 5)\) and \((4, 0)\). Identify both intercepts and explain their meaning in context.
  • ThinkThe y-intercept has \(x = 0\). The x-intercept has \(y = 0\).
  • Solve y-intercept: (0, 5) — the quantity starts at 5 when x = 0 x-intercept: (4, 0) — the quantity reaches zero when x = 4
  • Check\((0, 5)\) has \(x = 0\) ✓ · \((4, 0)\) has \(y = 0\) ✓
Answer: y-intercept \((0, 5)\) — start value. x-intercept \((4, 0)\) — endpoint.
Example 6.6.2
Finding intercepts from an equation
  • ProblemFind the x-intercept and y-intercept of \(2x + 3y = 12\).
  • ThinkSubstitute zero for each variable in turn to isolate the other.
  • Solve y-intercept (set x = 0): 2(0) + 3y = 12 → 3y = 12 → y = 4 y-intercept: (0, 4) x-intercept (set y = 0): 2x + 3(0) = 12 → 2x = 12 → x = 6 x-intercept: (6, 0)
  • Check\((0, 4)\): \(2(0) + 3(4) = 12\) ✓  ·  \((6, 0)\): \(2(6) + 3(0) = 12\) ✓
Answer: y-intercept at (0, 4); x-intercept at (6, 0).
Student Warning

Students often mix up intercepts. The y-intercept has \(x = 0\) and is a point on the y-axis. The x-intercept has \(y = 0\) and is a point on the x-axis. If you find a y-intercept with a nonzero x-value, you have made an error.

Section 6.7

Change in x and Change in y

Before understanding slope, you need to measure change between two points. Mathematicians use the Greek letter delta, \(\Delta\), to mean "change in." So \(\Delta x\) means "change in x" and \(\Delta y\) means "change in y." Using two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[ \Delta x = x_2 - x_1 \qquad \text{(change in x, horizontal)} \] \[ \Delta y = y_2 - y_1 \qquad \text{(change in y, vertical)} \]
Example 6.7.1
Calculating change between two points
  • ProblemFind \(\Delta x\) and \(\Delta y\) from \(P_1 = (2, 3)\) to \(P_2 = (7, 9)\).
  • ThinkSubtract the first coordinates from the second. Use the same order (second minus first) for both.
  • Solve Δx = x₂ − x₁ = 7 − 2 = 5 Δy = y₂ − y₁ = 9 − 3 = 6
  • CheckStarting at \((2, 3)\), add 5 to x and 6 to y: \((7, 9)\). ✓
Answer: \(\Delta x = 5\), \(\Delta y = 6\). Movement: 5 right and 6 up.
Example 6.7.2
Change with negative coordinates
  • ProblemFind \(\Delta x\) and \(\Delta y\) from \(A = (5, 8)\) to \(B = (1, 2)\).
  • Solve Δx = 1 − 5 = −4 Δy = 2 − 8 = −6
Answer: \(\Delta x = -4\), \(\Delta y = -6\). Movement: 4 left and 6 down.
Tip — Order Consistency

It does not matter which point you call \(P_1\) and which you call \(P_2\), as long as you are consistent. Whichever point is first in \(\Delta y\) must be first in \(\Delta x\). Mixing the order causes a sign error in the slope calculation.

Practice 6.6 – 6.7Intercepts and Change
  1. Find both intercepts of \(y = 3x + 6\).
  2. Find both intercepts of \(4x + y = 12\).
  3. Find both intercepts of \(3x - 5y = 15\).
  4. Find both intercepts of \(y = -4x\).
  5. \(\Delta x\) and \(\Delta y\) from \((1,3)\) to \((6,8)\)
  6. \(\Delta x\) and \(\Delta y\) from \((-2, 5)\) to \((4, -1)\)
  7. \(\Delta x\) and \(\Delta y\) from \((3, -4)\) to \((3, 9)\)
  8. \(\Delta x\) and \(\Delta y\) from \((0, 0)\) to \((-5, -3)\)
Sections 6.8 – 6.9
Slope as Rate of Change
The Slope Formula · Computing Slope · Four Types of Slope · Steepness
Section 6.8

Slope as Rate of Change

Slope is the ratio of vertical change to horizontal change between any two points on a line. Slope measures rate of change — it answers: for every 1 unit increase in x, how much does y change? The conventional symbol for slope is m.

\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \]

If \(m = 3\), every time x increases by 1, y increases by 3. If \(m = -2\), every time x increases by 1, y decreases by 2. Slope always has a unit and a meaning: in a distance-time graph it is speed; in a cost-quantity graph it is price per unit; in a temperature-time graph it is rate of heating or cooling.

Example 6.8.1
Finding slope between two points
  • ProblemFind the slope of the line through \((2, 3)\) and \((6, 11)\).
  • ThinkLabel points and apply the slope formula. Expect a positive result since y increases with x.
  • Solve m = (y₂ − y₁) / (x₂ − x₁) = (11 − 3) / (6 − 2) = 8 / 4 = 2
  • CheckMove from \((2,3)\) right 4 and up 8: \((6, 11)\). ✓
Answer: m = 2. For every 1 unit increase in x, y increases by 2 units.
Example 6.8.2
Negative slope
  • ProblemFind the slope of the line through \((1, 8)\) and \((5, 2)\).
  • Solve m = (2 − 8) / (5 − 1) = −6 / 4 = −3/2
  • CheckStart at \((1, 8)\), move right 4, down 6: \((5, 2)\). ✓
Answer: m = −3/2. For every 1 unit increase in x, y decreases by 1.5 units.
Example 6.8.3
Slope with negative coordinates
  • ProblemFind the slope of the line through \((-3, 1)\) and \((2, -4)\).
  • Solve m = (−4 − 1) / (2 − (−3)) = −5 / (2 + 3) = −5 / 5 = −1
  • CheckFrom \((-3,1)\), right 5 and down 5: \((2, -4)\). ✓
Answer: m = −1.
Student Warning — Subtracting a Negative

When a coordinate is negative, watch your subtraction carefully. Subtracting a negative means adding: \(2 - (-3) = 2 + 3 = 5\). Many slope errors come from this single step.

Slope is Constant on a Line

On a linear relationship, the slope is the same between any two points you choose. If you calculate different slopes between different pairs of points on the same graph, the relationship is not linear.

Section 6.9

Positive, Negative, Zero, and Undefined Slope

TypeVisualFormulaExample in Context
PositiveRises left to right\(m > 0\)Savings balance growing over time
NegativeFalls left to right\(m < 0\)Car loan balance decreasing with payments
ZeroHorizontal line\(m = 0\)Fixed price regardless of quantity
UndefinedVertical line\(\Delta x = 0\)An instantaneous boundary in geometry

A horizontal line has \(\Delta y = 0\), so \(m = 0/\Delta x = 0\). A vertical line has \(\Delta x = 0\), making the slope formula divide by zero — which is undefined. Zero slope and undefined slope are two completely different mathematical situations, not the same thing.

The absolute value of the slope determines steepness. A slope of \(m = 4\) is steeper than \(m = 1\) because the line rises more sharply. A slope of \(m = \frac{1}{4}\) is shallower.

Example 6.9.1
Classifying slope from two points
  • ProblemClassify slope for: (a) through \((2,3)\) and \((8,7)\)  (b) through \((1,5)\) and \((6,5)\)  (c) through \((4,9)\) and \((4,2)\)  (d) through \((0,10)\) and \((5,3)\)
  • Solve (a) m = (7−3)/(8−2) = 4/6 = 2/3 → Positive slope; rises left to right (b) m = (5−5)/(6−1) = 0/5 = 0 → Zero slope; horizontal line (c) m = (2−9)/(4−4) = −7/0 → Undefined slope; vertical line (d) m = (3−10)/(5−0) = −7/5 → Negative slope; falls left to right
Diagram Instruction

Draw four separate small coordinate planes side by side. In the first, draw a line with positive slope rising left to right, labeled "Positive slope: m > 0." In the second, draw a horizontal line, labeled "Zero slope: m = 0." In the third, draw a vertical line, labeled "Undefined slope." In the fourth, draw a line falling left to right, labeled "Negative slope: m < 0."

Practice 6.8 – 6.9Slope
  1. \((1,3)\) and \((4,9)\)
  2. \((2,7)\) and \((5,1)\)
  3. \((0,4)\) and \((6,4)\)
  4. \((3,5)\) and \((3,-2)\)
  5. \((-2,1)\) and \((4,7)\)
  6. \((-5,3)\) and \((1,-3)\)
  7. \((-1,-4)\) and \((3,-4)\)
  8. \((0,0)\) and \((5,10)\)

For each pair, compute the slope and classify it as positive, negative, zero, or undefined.

Sections 6.10 – 6.12
Building and Reading Graphs
Graphs from Tables · Graphs from Verbal Descriptions · Real-World Data and Scatter Plots
Section 6.10

Graphs from Tables

Procedure: Graphing from a Table

Step 1 — Read the table and identify x-values and y-values.

Step 2 — Write each row as an ordered pair.

Step 3 — Plot each ordered pair on the coordinate plane.

Step 4 — If the relationship is linear, draw a straight line through the points. If not, connect with a smooth curve or leave as individual points depending on context.

Step 5 — Label the axes with variable names and units.

Example 6.10.1
Building a graph from a table — delivery charges
  • ProblemA delivery driver charges a base fee of $5 plus $2 per mile. Build a table for \(x = 0, 1, 2, 3, 5, 10\) and graph the relationship.
  • ThinkTotal charge is \(y = 5 + 2x\). Substitute each x-value to find y.
  • Solve x = 0: y = 5 + 2(0) = 5 → (0, 5) x = 1: y = 5 + 2(1) = 7 → (1, 7) x = 2: y = 5 + 2(2) = 9 → (2, 9) x = 3: y = 5 + 2(3) = 11 → (3, 11) x = 5: y = 5 + 2(5) = 15 → (5, 15) x = 10: y = 5 + 2(10) = 25 → (10, 25)
  • CheckFor \(x = 5\): \(5 + 10 = 15\). ✓ Points fall in a straight line. ✓
Answer: Six points plotted. Straight line. y-intercept (0, 5) = base fee; slope 2 = $2 per mile.
Diagram Instruction

Draw a coordinate plane with x-axis labeled "Miles driven" from 0 to 10 and y-axis labeled "Total charge (dollars)" from 0 to 30. Plot and label each of the six points. Draw a straight line through them. Mark the y-intercept at (0, 5) and label it "Base fee: $5."

Example 6.10.2
Graph from a table — temperature by elevation
  • ProblemTemperature readings at different elevations: \((0, 20),\ (1, 14),\ (2, 8),\ (3, 2),\ (4, -4)\). Plot the data and interpret.
  • ThinkThe y-values decrease by 6 for each 1-unit increase in x. Linear relationship, negative slope.
  • Solve m = (14 − 20) / (1 − 0) = −6/1 = −6
Answer: Temperature drops 6°C for every km of elevation gain. y-intercept (0, 20) = temperature at ground level.
Section 6.11

Graphs from Verbal Descriptions

Procedure: Graphing from a Description

Step 1 — Identify the two quantities. Decide which is x and which is y.

Step 2 — Identify any starting value. This often becomes the y-intercept.

Step 3 — Identify how the relationship changes: constant, increasing, decreasing?

Step 4 — Identify any ending condition or special feature.

Step 5 — Plot key points and connect them in a way that reflects the described behavior.

Example 6.11.1
Steady decrease — draining water tank
  • ProblemA 24-gallon tank drains at a constant rate of 3 gallons per hour. Sketch a graph of volume remaining versus time.
  • ThinkVolume starts at 24 when \(t = 0\). It decreases by 3 per hour. It reaches zero after \(24 \div 3 = 8\) hours.
  • Solve Key points: Start: (0, 24) — tank is full Half: (4, 12) — half full End: (8, 0) — tank is empty Slope: m = −3
Answer: Draw a straight line from (0, 24) to (8, 0). y-intercept = starting volume; x-intercept = when tank empties; slope = −3 gallons/hour.
Diagram Instruction

Draw a coordinate plane with x-axis labeled "Time (hours)" from 0 to 10 and y-axis labeled "Volume (gallons)" from 0 to 30. Plot (0, 24), (4, 12), and (8, 0). Draw a straight line through them. Label the y-intercept at (0, 24) and the x-intercept at (8, 0).

Example 6.11.2
Piecewise description — runner's speed
  • ProblemA runner runs at 6 mph for 3 hours, rests for 1 hour, then runs at 4 mph for 2 more hours. Sketch a graph of speed over time.
  • ThinkSpeed is constant during each segment. Three horizontal sections at different heights.
  • Solve Time 0 to 3 hours: speed = 6 mph (horizontal segment at y = 6) Time 3 to 4 hours: speed = 0 mph (horizontal segment at y = 0) Time 4 to 6 hours: speed = 4 mph (horizontal segment at y = 4)
Answer: Three separate horizontal segments. This piecewise graph shows that speed can hold constant, then change abruptly.
Reading the Language of Graphs

When a description says "constant" or "steady," the graph segment is flat (zero slope). When it says "increasing at a constant rate" or "decreasing at a constant rate," the segment is a straight line with nonzero slope.

Section 6.12

Graphing Real-World Data

Real data is messier than equations: points do not always fall perfectly on a line, and some variation is expected. When real data is graphed with each point plotted individually and no line connecting them, the result is called a scatter plot. Scatter plots are used to look for patterns, trends, and associations.

If the points rise from left to right overall, there is a positive association. If the points fall from left to right, there is a negative association. If the points show no clear direction, there is no association. You will study correlation and lines of best fit formally in Chapter 14.

Example 6.12.1
Graphing a data set and describing its pattern
  • ProblemPlot and describe: hours studied versus test score — \((1, 62),\ (2, 71),\ (3, 78),\ (4, 85),\ (5, 90)\).
  • ThinkEach row is an ordered pair. Plot them and look for direction and shape.
  • SolvePoints plotted as a scatter plot. Points rise left to right. Approximately linear.
  • CheckFrom \((1,62)\) to \((5,90)\): \(\Delta y = 28\), \(\Delta x = 4\), slope ≈ 7. Each hour of study adds about 7 points. ✓
Answer: Positive association. Each additional hour of study is associated with roughly 7–8 more score points.
Diagram Instruction

Draw a coordinate plane with x-axis labeled "Hours studied" from 0 to 6 and y-axis labeled "Test score" from 50 to 100. Plot each of the five points as individual dots. Do not connect them with a line.

Practice 6.10 – 6.12Building and Reading Graphs
  1. A plumber charges $40 base plus $60 per hour. (a) Create a table for \(h = 0, 1, 2, 3, 4\). (b) Graph the relationship. (c) Identify and interpret the slope and y-intercept.
  2. A car's 14-gallon tank uses 0.04 gallons per mile. (a) Write an equation for gallons remaining G after d miles. (b) Create a table for \(d = 0, 50, 100, 200, 350\). (c) Graph and identify the x-intercept.
  3. Temperature over a morning: \((0, 14),\ (1, 17),\ (2, 22),\ (3, 26),\ (4, 31)\). (a) Plot the data. (b) Is the relationship approximately linear? (c) Estimate the slope and interpret it.
  4. Describe a real situation that would produce a graph with a negative slope. Identify what the intercepts would represent in your situation.
  5. Describe the shape of a graph for this situation: a car accelerates steadily for 5 seconds, then holds a constant speed for 10 seconds, then brakes to a stop over 3 seconds. Sketch the speed-versus-time graph.
Sections 6.13 – 6.14
Common Mistakes and the Road Ahead
Six Recurring Errors · Prevention Strategies · Bridge to Linear Equations
Section 6.13

Common Coordinate and Slope Mistakes

Mistake 1 — Reversing x and y in an Ordered Pair
"3 right and 5 up" → written as (5, 3)

In an ordered pair \((x, y)\), x comes first. x is horizontal, y is vertical. Always write horizontal movement first as the x-coordinate, vertical second as y.

Mistake 2 — Subtracting Coordinates in the Wrong Order
Points (2,5) and (6,9): Δy = 5−9 = −4 but Δx = 6−2 = 4 → slope = −1 (wrong)

The order of subtraction must be the same for both Δy and Δx. Label points as \((x_1,y_1)\) and \((x_2,y_2)\) and use \(m = (y_2 - y_1)/(x_2 - x_1)\) consistently.

Mistake 3 — Confusing Rise and Run
slope = run/rise = Δx/Δy

Slope is rise over run: \(m = \Delta y / \Delta x\). The vertical change (rise) goes on top; horizontal change (run) goes on the bottom.

Mistake 4 — Treating Zero Slope and Undefined Slope as the Same
"Horizontal and vertical lines both have no slope."

Zero slope means the slope formula gives exactly 0 (horizontal line). Undefined slope means the formula requires division by zero (vertical line). These are two completely different mathematical situations.

Mistake 5 — Misidentifying Intercepts
Student calls (3, 0) the y-intercept because "y is the second coordinate."

The y-intercept is on the y-axis: \(x = 0\). The x-intercept is on the x-axis: \(y = 0\). Check the zero — whichever coordinate is zero tells you which intercept it is.

Mistake 6 — Writing Slope as a Bare Number Without Meaning
In an application: "Slope = 4." (no units, no interpretation)

In context, slope is always a rate of change. If x is hours and y is dollars, then slope 4 means $4 per hour. Always interpret slope in the language of the problem.

Section 6.14

Preparing for Linear Equations

You have now built a strong visual and conceptual foundation for Chapter 7. Everything you practiced in this chapter connects directly to the formal study of linear equations.

Chapter 6 SkillWhere It Appears in Chapter 7
The coordinate planeSurface on which every equation and function is graphed
Ordered pairsSolutions to equations; input-output records of functions
y-interceptThe constant b in slope-intercept form \(y = mx + b\)
SlopeThe coefficient m in \(y = mx + b\); also in point-slope form
Graphs from tablesVerification that an equation and its graph agree
Graphs from descriptionsAlgebraic modeling: translating real situations into equations

In Chapter 7, you will write equations for lines, not just draw or read them. You will move fluently among equations, graphs, tables, and verbal descriptions, using each representation to understand the others.

Preview of Chapter 7

The equation \(y = mx + b\) is called slope-intercept form. You already understand what m and b represent: m is the slope (rate of change), and b is the y-intercept (starting value). Chapter 7 will teach you how to write this equation from a graph, from a table, from two points, and from a real-world description. You will also learn point-slope form and standard form.

Chapter 6 — Back Matter

Chapter Summary

Key Terms Review

coordinate plane
A flat surface formed by two perpendicular number lines, used for graphing ordered pairs and functions.
x-axis
The horizontal number line in a coordinate plane.
y-axis
The vertical number line in a coordinate plane.
origin
The point where the x-axis and y-axis intersect, written as (0, 0).
ordered pair
A pair of numbers written as (x, y) that identifies a location on the coordinate plane. Order is fixed: x first, y second.
quadrant
One of the four regions of the coordinate plane, labeled I through IV, each with a distinct sign pattern for x and y.
x-intercept
The point where a graph crosses the x-axis. At this point, y = 0. Written as (a, 0).
y-intercept
The point where a graph crosses the y-axis. At this point, x = 0. Written as (0, b).
rise
The vertical change between two points; \(\Delta y = y_2 - y_1\).
run
The horizontal change between two points; \(\Delta x = x_2 - x_1\).
slope
The ratio of vertical change to horizontal change: \(m = \Delta y / \Delta x\). Slope measures rate of change.
rate of change
How much one quantity changes for each unit change in another quantity. Slope gives the rate of change of y per unit of x.
positive slope
A line that rises from left to right; m > 0.
negative slope
A line that falls from left to right; m < 0.
zero slope
A horizontal line; \(\Delta y = 0\), so m = 0. The y-value does not change.
undefined slope
A vertical line; \(\Delta x = 0\), making the slope formula undefined. Not the same as zero slope.
scatter plot
A graph of real data points plotted individually, without a connecting line, used to identify patterns and associations.
positive association
A pattern in a scatter plot where y tends to increase as x increases.

Concept Review Questions

Answer each question in full sentences.

  1. What two number lines form the coordinate plane, and how are they oriented relative to each other?
  2. What is the origin, and what ordered pair represents it?
  3. Describe the sign pattern for each of the four quadrants.
  4. What is the difference between the x-intercept and the y-intercept? How are they found from a graph? How are they found from an equation?
  5. What does slope measure? Why is slope described as a rate of change?
  6. Explain in words what it means for a line to have slope \(m = -3\).
  7. Why is the slope of a vertical line undefined rather than zero?
  8. A set of data points all fall exactly on a straight line. What does this tell you about the relationship between x and y?
  9. How can you tell from a scatter plot whether an association is positive or negative?
  10. A graph shows a flat segment from \((2, 8)\) to \((7, 8)\), then a falling segment to \((10, 2)\). Describe in words what this graph communicates.

Skill Practice

Part ACoordinate Plane

Identify the quadrant or axis location for each ordered pair.

  1. \((5, 2)\)
  2. \((-3, 7)\)
  3. \((-1, -8)\)
  4. \((4, -6)\)
  5. \((0, 5)\)
  6. \((-7, 0)\)
  7. \((0, 0)\)
  8. \((2, -2)\)
Part BIntercepts

Find the x-intercept and y-intercept of each equation.

  1. \(y = 3x + 6\)
  2. \(y = -2x + 8\)
  3. \(4x + y = 12\)
  4. \(3x - 5y = 15\)
  5. \(y = \frac{1}{2}x - 3\)
  6. \(2x + 7y = 14\)
  7. \(x - y = 9\)
  8. \(y = -4x\)
Part CSlope

Find the slope between each pair of points. Classify as positive, negative, zero, or undefined.

  1. \((1, 3)\) and \((4, 9)\)
  2. \((2, 7)\) and \((5, 1)\)
  3. \((0, 4)\) and \((6, 4)\)
  4. \((3, 5)\) and \((3, -2)\)
  5. \((-2, 1)\) and \((4, 7)\)
  6. \((-5, 3)\) and \((1, -3)\)
  7. \((-1, -4)\) and \((3, -4)\)
  8. \((0, 0)\) and \((5, 10)\)

Word Problems

For each problem: define variables, write an equation or expression when needed, create a table, graph the relationship, and interpret the slope and intercepts.

  1. A plumber charges a flat fee of $40 plus $60 per hour. (a) Write an equation for total cost C after h hours. (b) Create a table for \(h = 0, 1, 2, 3, 4\). (c) Graph. (d) Interpret the slope and y-intercept.
  2. A car's gas tank holds 14 gallons. The car uses 0.04 gallons per mile. (a) Write an equation for gallons remaining G after d miles. (b) Create a table for \(d = 0, 50, 100, 200, 350\). (c) Graph. (d) What is the x-intercept and what does it mean?
  3. The table shows outdoor temperature over several winter morning hours: \(x = 0\) (6 AM) through \(x = 4\) (10 AM), with temperatures 14, 17, 22, 26, 31°F. (a) Plot the data. (b) Is the relationship approximately linear? (c) Estimate and interpret the slope.
  4. A student saves $15 per week starting from a balance of $0. (a) Complete a table for weeks 0 through 6. (b) Graph. (c) After how many weeks will the student have $90? Show this on the graph.
  5. A line passes through \((0, 10)\) and \((5, 0)\). If x = hours and y = liters of water in a container, find the slope, both intercepts, and interpret each in context.
  6. A baker starts with 30 cups of flour and uses 3 cups per loaf. (a) Write an equation for flour remaining F after n loaves. (b) Graph. (c) Interpret the slope and x-intercept.

Error Analysis

Each solution contains an error. Identify the mistake and write the correct solution.

Problem 1
Slope between (3,7) and (8,2): m = (8−3)/(2−7) = 5/(−5) = −1

Identify the error and find the correct slope.

Problem 2
The point (0, 4) is the x-intercept of y = 2x + 4.

Identify the error. What type of intercept is (0, 4) actually?

Problem 3
The line through (5, 3) and (5, −1) has slope 0.

Identify the error and give the correct classification.

Problem 4
"A line with slope −1/3 falls steeply because the slope is a small fraction."

Explain why this reasoning is incorrect. What does the absolute value of slope actually tell you?

Problem 5
A student draws a graph with x-axis labeled "Cost (dollars)" and y-axis labeled "Number of items," then describes the y-intercept as "the cost when items equal zero."

Explain the error. What does the y-intercept actually represent in this labeling?

Chapter 6 Checkpoint

Chapter 6 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 6
Part A — Coordinate Plane
  1. Identify the quadrant for each point: \((6, -2)\), \((-4, 5)\), \((-3, -7)\).
  2. Plot the following points on a coordinate plane: \((0, 4)\), \((3, -2)\), \((-5, 1)\), \((2, 0)\).
Part B — Intercepts
  1. Find both intercepts of \(5x - 2y = 10\).
  2. Find both intercepts of \(y = -3x + 9\).
Part C — Slope
  1. Find the slope between \((2, -1)\) and \((8, 5)\).
  2. Find the slope between \((-3, 4)\) and \((-3, -6)\). Classify it.
  3. Find the slope between \((0, 7)\) and \((5, 7)\). Classify it.
Part D — Graphs from Tables
  1. The rule gives \(y = 4x - 2\). Create a table for \(x = -1, 0, 1, 2, 3\) and graph the result. Identify the slope and y-intercept from your graph.
Part E — Graph Interpretation
  1. A line passes through \((0, 10)\) and \((5, 0)\). Find the slope, x-intercept, and y-intercept. If x is hours and y is liters of water, interpret each value in context.
Part F — Real-World Application
  1. A baker uses 3 cups of flour per loaf and starts with 30 cups.
    (a) Write an equation for flour remaining \(F\) after baking \(n\) loaves.
    (b) Graph the equation.
    (c) What is the slope? What does it mean?
    (d) What is the x-intercept? What does it represent?

Answers may be placed in the answer key.

Bridge Note
Chapter 7: Linear Functions and Equations

Everything in this chapter was preparation for the formal study of linear functions and equations. In Chapter 7, you will take the slope and intercepts you found here and build them into the equations of lines. The central form is \(y = mx + b\), where m is the slope and b is the y-intercept — both values you calculated throughout this chapter. You will write equations from graphs, from two points, and from real-world descriptions. You will also learn point-slope form, useful when you know one point and the slope, and standard form, which places both x and y on the same side of the equation. By the end of Chapter 7, you will move freely among equations, graphs, tables, and verbal descriptions. The coordinate plane skills you built here are the foundation for all of that fluency.

MGU Library Connection
  • Appendix H: Linear Equation Forms — slope-intercept, point-slope, standard form reference
  • Appendix E: Equation-Solving Strategy Guide — solving for intercepts and slope
  • MGU Physics supplement: Graphing Motion — slope as velocity, intercepts as initial conditions
  • MGU Economics supplement: Supply and Demand Graphs — slope and intercepts in economic models
  • Chapter 14: Data, Statistics, and Algebraic Modeling — scatter plots, correlation, and lines of best fit