Chapter 7 — Graphing Lines

7.1 Learning Objectives

By the end of this chapter, you should be able to:

understand what graphs represent

interpret coordinate systems

plot ordered pairs correctly

understand linear graphs visually

identify slope from graphs and equations

identify intercepts

graph equations in slope-intercept form

understand horizontal and vertical lines

connect equations to geometric structure

interpret graphs as models of changing systems

7.2 Big Picture — Graphs Turn Equations Into Pictures

Equations describe relationships symbolically.

Graphs make those relationships visible.

Example:

y = 2x + 1

This equation describes:

how y changes when x changes

Its graph visually reveals:

direction

growth

steepness

starting position

behavior over time

Graphs allow humans to SEE mathematics.

This is one of the most powerful ideas in algebra.

7.3 The Coordinate Plane

Graphs are built on the coordinate plane.

The coordinate plane contains:

a horizontal axis

a vertical axis

Horizontal Axis

The horizontal axis is called:

x-axis

Vertical Axis

The vertical axis is called:

y-axis

Origin

The point where the axes meet is called:

origin

Location:

(0,0)

7.4 Ordered Pairs

Points are written as ordered pairs.

Example:

(3, 2)

means:

move right 3

move up 2

The order matters.

Example:

(2, 3)

is a DIFFERENT point.

7.5 Quadrants

The coordinate plane is divided into four regions called quadrants.

Quadrant x y

I positive positive

II negative positive

III negative negative

IV positive negative

7.6 Plotting Points

Example:

(-2, 4)

Move:

left 2

up 4

Example:

(5, -3)

Move:

right 5

down 3

Plotting points creates visual representations of relationships.

7.7 Linear Equations Produce Straight Lines

Linear equations graph as straight lines.

Example:

y = x

produces a diagonal line.

Example:

y = 2x

produces a steeper line.

Straight lines represent:

constant rates of change

predictable behavior

proportional relationships

7.8 Slope — The Rate of Change

Slope describes:

steepness

direction

growth rate

In:

y = mx + b

the value:

m

is the slope.

7.9 Positive Slope

Example:

y = 2x + 1

As:

x increases

y increases

The line rises left-to-right.

7.10 Negative Slope

Example:

y = -x + 3

As:

x increases

y decreases

The line falls left-to-right.

7.11 Zero Slope

Example:

y = 4

This creates a horizontal line.

The y-value never changes.

7.12 Undefined Slope

Example:

x = 3

This creates a vertical line.

Vertical lines have:

undefined slope

because division by zero would occur.

7.13 Slope Formula

Given two points:

(x₁, y₁)

(x₂, y₂)

Slope is:

m = (y₂ - y₁)/(x₂ - x₁)

This measures:

rise/run

7.14 Example of Slope Formula

Points:

(1,2)

(3,6)

Compute:

m = (6 - 2)/(3 - 1)

Simplify:

m = 4/2 = 2

7.15 Slope-Intercept Form

The most common graphing form is:

y = mx + b

Where:

m = slope

b = y-intercept

7.16 The y-Intercept

The y-intercept is:

where the graph crosses the y-axis

Example:

y = 3x + 2

The intercept is:

(0,2)

7.17 Graphing Using Slope and Intercept

Example:

y = 2x + 1

Step 1:

plot intercept:

(0,1)

Step 2:

use slope:

rise 2

run 1

Step 3:

draw line through points.

7.18 Real-World Graphs

Graphs model:

motion

economics

population growth

electrical systems

networking

computing performance

financial trends

Graphs allow humans to interpret changing systems visually.

7.19 Common Beginner Difficulties

Students often struggle with:

switching x and y

graph direction

slope signs

plotting errors

confusion about rise/run

interpreting intercepts

These difficulties are normal.

Graph intuition develops through:

repetition

visualization

experimentation

observation

7.20 Mental Model

Equations are symbolic descriptions.

Graphs are visual descriptions.

A graph is essentially:

a picture of an equation’s behavior

7.21 Warm-Up Problems

Problems

Plot:

(2,3)

Plot:

(-1,4)

Plot:

(5,-2)

Identify the quadrant:

(-3,2)

Identify the quadrant:

(4,-5)

Identify the slope in:

y = 3x + 2

Identify the slope in:

y = -2x + 1

Identify the intercept in:

y = 5x - 4

Identify the intercept in:

y = -x + 7

State whether slope is:

positive

negative

zero

undefined

for:

y = 6

State whether slope is:

positive

negative

zero

undefined

for:

x = -3

State whether slope is:

positive

negative

zero

undefined

for:

y = -4x + 2

7.22 Guided Problems

Problems

Find slope between:

(1,2)

(3,6)

Find slope between:

(2,5)

(6,5)

Find slope between:

(4,1)

(4,7)

Graph conceptually:

y = x

Graph conceptually:

y = -x

Identify slope and intercept:

y = 4x - 3

Identify slope and intercept:

y = -2x + 5

Write slope-intercept form for:

slope = 3

intercept = 2

Write slope-intercept form for:

slope = -1

intercept = 4

Describe the graph of:

y = 2

Describe the graph of:

x = 5

Explain what slope measures.

7.23 Challenge Problems

Find slope between:

(-2,3)

(4,-9)

Find slope between:

(1,-1)

(5,7)

Determine whether the following line rises or falls:

y = -3x + 1

Determine whether the following line rises or falls:

y = 5x - 7

Write a linear equation with:

slope = 2

intercept = -5

Write a linear equation with:

slope = -4

intercept = 3

Explain why vertical lines have undefined slope.

Explain why horizontal lines have zero slope.

Describe a real-world system that could be represented graphically.

Explain why graphing is important in science and engineering.

7.24 Solutions

Solutions to Warm-Up Problems

right 2, up 3

left 1, up 4

right 5, down 2

Quadrant II

Quadrant IV

3

-2

-4

7

zero slope

undefined slope

negative slope

Solutions to Guided Problems

2

0

undefined

Positive diagonal line through origin.

Negative diagonal line through origin.

slope = 4

intercept = -3

slope = -2

intercept = 5

y = 3x + 2

y = -x + 4

Horizontal line crossing y-axis at 2.

Vertical line crossing x-axis at 5.

Slope measures rate of change and steepness.

Solutions to Challenge Problems

-2

2

Falls left-to-right.

Rises left-to-right.

y = 2x - 5

y = -4x + 3

Vertical lines would require division by zero in the slope formula.

Horizontal lines have no vertical change, so rise equals zero.

Possible examples include:

population growth

financial trends

temperature changes

speed vs time

network throughput

electrical behavior

Graphs provide visual understanding of changing systems and help humans interpret complex relationships quickly.