The Coordinate Plane

Chapter 1: Variables, Expressions and Integers

Lesson 1.8: Graphing in a Coordinate Plane

Using ordered pairs to locate points, describe patterns, and solve problems

Learning Objectives

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

Identify and plot points using ordered pairs in a coordinate plane.
Use coordinate graphs to describe patterns and solve geometric and real-life problems.

God Brings Order and Purpose

A coordinate plane gives us an orderly way to describe where a point is located. In the same way, God is a God of order and purpose. When we organise information carefully, we are better able to recognise patterns, make wise decisions, and appreciate the beauty of creation.

Memory verse: “But everything should be done in a fitting and orderly way.” — 1 Corinthians 14:40

1. The Coordinate Plane

A coordinate plane is formed by two number lines that intersect at right angles:

The horizontal number line is the x-axis.
The vertical number line is the y-axis.
The axes intersect at the origin, written as $(0,0)$ .

The axes divide the coordinate plane into four regions called quadrants.

Signs in the four quadrants:

Quadrant I: (+,+) Quadrant II: (-,+)

Quadrant III: (-,-) Quadrant IV: (+,-)

Picture space for a coordinate plane, axes, origin, and quadrants — The x-axis is horizontal, the y-axis is vertical, and the origin is $(0,0)$ .

2. Ordered Pairs

Each point in a coordinate plane is represented by an ordered pair, written as $(x,y)$ .

The first number is the x-coordinate. It tells how far to move left or right.
The second number is the y-coordinate. It tells how far to move up or down.

Worked Example 1: Naming a Point

Point $P$ is 3 units left of the origin and 2 units down.

$P=(-3,-2)$

Because both coordinates are negative, point $P$ is in Quadrant III.

Worked Example 2: Plotting a Point

To plot $A(4,1)$ , begin at the origin. Move 4 units right because the x-coordinate is 4. Then move 1 unit up because the y-coordinate is 1. Point $A$ lies in Quadrant I.

Important: Always move horizontally first and vertically second.

3. Points on the Axes

A point is on an axis when one of its coordinates is zero.

$B(0,-3)$ lies on the y-axis.

$C(4,0)$ lies on the x-axis.

$(0,0)$ is the origin.

Picture space for plotting ordered pairs in four quadrants — Read and plot the x-coordinate first, then the y-coordinate.

4. Graphing an Equation

A graph can show all the ordered pairs that make an equation true. For example, consider:

$y=5-x$

Choose values for $x$ , calculate the matching value of $y$ , and plot the ordered pairs.

$x$	$y=5-x$	Ordered pair
0	5	$(0,5)$
1	4	$(1,4)$
2	3	$(2,3)$
3	2	$(3,2)$

These points lie on a straight line. As $x$ increases by 1, $y$ decreases by 1.

5. Scatter Plots and Relationships

A scatter plot is a graph that displays pairs of data. Each data pair is plotted as a point $(x,y)$ .

After making a scatter plot, look for a relationship:

An upward pattern suggests that $y$ tends to increase as $x$ increases.
A downward pattern suggests that $y$ tends to decrease as $x$ increases.
Points with no clear pattern suggest little or no relationship.

Common Mistakes to Avoid

Do not reverse the coordinates. $(x,y)$ is not the same as $(y,x)$ .
Do not move vertically before moving horizontally.
Remember that a point with an x-coordinate of 0 is on the y-axis.
Remember that a point with a y-coordinate of 0 is on the x-axis.

Practice Questions

Use graph paper or a coordinate grid. Label each point carefully and show your working.

1. Skill Practice

A. Name the quadrant or axis where each point is located.

$A(8,6)$
$B(-8,6)$
$C(-4,-11)$
$D(3,-6)$
$E(9,0)$
$F(-4,1)$
$G(-10,-10)$
$H(0,-8)$
$I(6,-2)$
$J(2,13)$

B. Make a table of values and graph six ordered pairs for $y=5-x$ .

Use $x=-2,-1,0,1,2,3$ . Describe the graph.

C. Plot each point and name its quadrant or axis.

$P(4,3)$
$Q(-5,2)$
$R(-2,-4)$
$S(6,-3)$
$T(0,5)$
$U(-7,0)$

D. Geometry on a Coordinate Plane

Plot $A(1,4)$ , $B(5,4)$ , $C(5,1)$ , and $D(1,1)$ . Connect the points in order.
What shape is formed? Explain your answer.
Subtract 3 from each coordinate of every vertex. Write the coordinates of the new figure.

2. Applications

Driving data: The table shows the average distance travelled per passenger car in the United States.

Years since 1980	0	5	10	15	20	25
Average distance (thousands of miles)	8.8	9.4	10.5	11.2	12.0	12.4

20a. Plot the data as a scatter plot.

20b. Does the scatter plot suggest a relationship between the year and the average distance travelled? Explain.

Fuel economy: The engine sizes of several cars and their average mileage are shown below.

Engine size (litres)	1	2	3	4	5	6
Mileage (miles per gallon)	47	33	28	25	22	19

21a. Plot the data as a scatter plot.

21b. Describe the relationship between engine size and mileage.

Weather graph: The daily temperatures over five days are represented by , , , , and , where the first coordinate is the day number and the second coordinate is the temperature in degrees Celsius.
1. Plot the points and join consecutive points.
2. Which day had the highest temperature?
3. Which day had the lowest temperature?
Geometry: Plot the points $(-3,-2)$ , $(-3,6)$ , $(5,6)$ , and $(5,-2)$ . Connect them in order and connect the last point to the first. Identify the figure and explain your reasoning.
Translation: Point $P(-4,3)$ is translated 5 units right and 2 units down. Write the coordinates of the image of $P$ .
Challenge: Find two different ordered pairs that lie on the line $y=2x+1$ . Explain how you know they lie on the line.

Answers

Show answers: Skill Practice

Quadrant I
Quadrant II
Quadrant III
Quadrant IV
x-axis
Quadrant II
Quadrant III
y-axis
Quadrant IV
Quadrant I
Table values: $(-2,7)$ , $(-1,6)$ , $(0,5)$ , $(1,4)$ , $(2,3)$ , and $(3,2)$ . The graph is a straight line that decreases as $x$ increases.
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
y-axis
x-axis
Plot the four vertices and connect them in order.
Rectangle
New vertices: $(-2,1)$ , $(2,1)$ , $(2,-2)$ , and $(-2,-2)$ .

Show answers: Applications

a. Plot $(0,8.8)$ , $(5,9.4)$ , $(10,10.5)$ , $(15,11.2)$ , $(20,12.0)$ , and $(25,12.4)$ . b. Yes. The upward pattern suggests that average distance travelled increased as the years increased.
a. Plot $(1,47)$ , $(2,33)$ , $(3,28)$ , $(4,25)$ , $(5,22)$ , and $(6,19)$ . b. The relationship is negative: as engine size increases, mileage generally decreases.
a. Plot and connect the five points. b. Day 3, at $4^\circ\text{C}$ . c. Day 1, at $-3^\circ\text{C}$ .
Square. Each side has length 8 units, and the four angles are right angles.
$P'(1,1)$ .
Answers may vary. For example, $(0,1)$ and $(1,3)$ both satisfy $y=2x+1$ .

Show the key ideas to remember

The x-coordinate is read first; the y-coordinate is read second.
Move horizontally first, then vertically.
The origin is $(0,0)$ .
A point with x-coordinate 0 lies on the y-axis.
A point with y-coordinate 0 lies on the x-axis.
Quadrant I is $(+,+)$ , Quadrant II is $(-,+)$ , Quadrant III is $(-,-)$ , and Quadrant IV is $(+,-)$ .
Scatter plots help us look for relationships between two quantities.

Chapter 1 Summary

In this chapter, you learned how to use variables and expressions, powers and exponents, the order of operations, integers, absolute value, integer operations, and coordinate planes. These ideas provide a strong foundation for solving equations and working with functions in the next chapter.

Remember 1 Corinthians 14:40: “Everything should be done in a fitting and orderly way.”

Lesson 1.8: Graphing in a Coordinate Plane

Learning Objectives

God Brings Order and Purpose

1. The Coordinate Plane