Graphing Relations | JHS Mathematics | The Let's Learn Project

Introduction

Graphing relations helps us visualize mathematical relationships between variables. In this lesson, we'll learn how to:

Locate points on the number plane (Cartesian plane)
Create tables of values for given relations
Draw graphs from these relations
Use graphs to solve problems

Key Terms

Number Plane (Cartesian Plane): A two-dimensional plane formed by two perpendicular number lines (axes)

Ordered Pair (x, y): A pair of numbers that gives the position of a point on the plane

Relation: A rule that connects input values (x) to output values (y)

Number Plane

Locating Points

The number plane has two axes:

x-axis: Horizontal number line
y-axis: Vertical number line

Points are located using ordered pairs (x, y):

Example: Point A(2, 3) means:

Move 2 units right along x-axis
Move 3 units up along y-axis

Quadrants

The plane is divided into 4 quadrants:

Quadrant	x-coordinate	y-coordinate
I	Positive (+)	Positive (+)
II	Negative (-)	Positive (+)
III	Negative (-)	Negative (-)
IV	Positive (+)	Negative (-)

Plotting Points

To plot a point (x, y):

Start at the origin (0,0)
Move horizontally to the x-coordinate
Move vertically to the y-coordinate
Mark the point with a dot

Example: Plot these points:

A(2, 3), B(-1, 4), C(0, -2), D(-3, -1)

Tables of Values

Creating Tables

For a given relation (like y = 2x + 1), we can create a table of values by:

Choosing x-values (usually -2, -1, 0, 1, 2)
Calculating the corresponding y-values
Recording the pairs in a table

Example: Create a table for y = 2x + 1

x	y = 2x + 1	(x, y)
-2	2(-2)+1 = -3	(-2, -3)
-1	2(-1)+1 = -1	(-1, -1)
0	2(0)+1 = 1	(0, 1)
1	2(1)+1 = 3	(1, 3)
2	2(2)+1 = 5	(2, 5)

Practice Example

Create a table of values for y = x² - 2

x	y = x² - 2	(x, y)
-2	(-2)²-2 = 2	(-2, 2)
-1	(-1)²-2 = -1	(-1, -1)
0	(0)²-2 = -2	(0, -2)
1	(1)²-2 = -1	(1, -1)
2	(2)²-2 = 2	(2, 2)

Drawing Graphs

Steps to Draw Graphs

To graph a relation:

Create a table of values
Draw and label the x and y axes
Plot all points from the table
Connect the points with a smooth line or curve

Linear Relations

Forms straight lines (e.g., y = 2x + 1)

Example: Graph of y = x + 2

Non-linear Relations

Forms curves (e.g., y = x²)

Example: Graph of y = x²

Graph Characteristics

Important features to note:

y-intercept: Where the graph crosses the y-axis (x=0)
x-intercept: Where the graph crosses the x-axis (y=0)
Slope: Steepness of a line (rise/run)
Vertex: Highest/lowest point on a parabola

Applications

Practical Examples

Graphing is useful in many real-world situations:

Distance-Time Graphs

Shows how distance changes with time:

Example: A car travels at constant speed of 60km/h

Relation: distance = 60 × time

Cost-Profit Analysis

Businesses use graphs to find break-even points:

Example: Fixed costs = \$200, Price per item = \$5

Profit = 5x - 200 (where x is number of items sold)

Temperature Changes

Tracking temperature over time:

Example: Hourly temperature readings throughout a day

Practice Exercise

Question 1

Plot these points on a number plane: A(2, 3), B(-1, 0), C(0, -4), D(-3, 2)

Draw axes and plot each point according to its coordinates

Question 2

Create a table of values for y = 3x - 2 (use x = -1, 0, 1, 2)

x	y = 3x - 2
-1	3(-1)-2 = -5
0	3(0)-2 = -2
1	3(1)-2 = 1
2	3(2)-2 = 4

Question 3

Graph the relation y = -x + 3 using x-values from -2 to 2

Table of values:

x	y = -x + 3
-2	-(-2)+3 = 5
-1	-(-1)+3 = 4
0	-(0)+3 = 3
1	-(1)+3 = 2
2	-(2)+3 = 1

Plot these points and connect with a straight line

Question 4

Find the x-intercept and y-intercept of y = 2x - 4

y-intercept (x=0): y = 2(0)-4 = -4 → (0, -4)

x-intercept (y=0): 0 = 2x-4 → 2x=4 → x=2 → (2, 0)

Question 5

A taxi charges \$3 base fare plus \$2 per km. Write the relation and graph it for 0 to 5 km.

Relation: Cost = 2 × distance + 3 or C = 2d + 3