Linear Functions, Equations and Inequalities

Lessons

Senior 1 · Mathematics

Linear Functions, Equations and Inequalities

5 lessons

Lesson 1 of 5

The number line and the Cartesian plane

22 min

Let's begin

A seat number in a stadium, a square on a draughts board and a grid reference on a map all do the same job: they tell you exactly where something is. Mathematics does this with numbers. One number fixes a point on a line. Two numbers fix a point on a flat surface. In this lesson you will learn the system that every graph in this unit is built on.

1. The number line

Definition

Number line: a straight line on which every point stands for one real number. A point called zero is chosen, a unit length is marked off equally in both directions, positive numbers go to the right and negative numbers to the left.

Example: on the line below, $- 1.5$ sits halfway between $- 2$ and $- 1$ .

Rule. On a number line, a number is greater than every number to its left. So $2 > - 3$ and $- 1.5 > - 3$ , even though 3 looks "bigger" than 1.5. The reason is that moving right always means adding a positive amount.

2. Two number lines make a plane

A single line cannot describe the position of a desk in a classroom: you need to know how far along the row and how far back. So we draw two number lines at right angles.

Definition

Cartesian plane (or coordinate plane): a flat surface on which two perpendicular number lines, the axes, cross at their zeros.

The horizontal line is the $x$ -axis; the vertical line is the $y$ -axis.
The point where they cross is the origin, $O (0, 0)$ .
The axes divide the plane into four quadrants, numbered I to IV anticlockwise from the top right.

Definition

Coordinates (an ordered pair): the pair of numbers $(x, y)$ that fixes a point. The first number, the $x$ -coordinate, is the distance across from the $y$ -axis. The second, the $y$ -coordinate, is the distance up or down from the $x$ -axis.

Notation: $P (3, 2)$ means "point $P$ has $x = 3$ and $y = 2$ ".

To reach $P (3, 2)$ , start at $O$ , move 3 units to the right along the $x$ -axis, then 2 units up. The dashed path in the diagram shows this.

Maps use the same idea. Longitude is measured east or west (across) and latitude north or south (up or down). Kigali is at about $30.1^\circ$ E, $1.9^\circ$ S. If east and north count as positive, Kigali sits near the point $(30.1, - 1.9)$ . Its $x$ is positive and its $y$ is negative, so it lies in quadrant IV.

3. Drawing and labelling axes

Draw the $x$ -axis across and the $y$ -axis up, crossing at right angles. Use a ruler.
Choose a scale, such as 1 cm for 1 unit, and mark it equally on both sides of $O$ . The two axes may use different scales, but each axis must keep its own scale.
Number the marks: positive to the right and up, negative to the left and down.
Write $x$ at the end of the horizontal axis, $y$ at the top of the vertical axis, and $O$ at the origin.

4. Signs in the four quadrants

Quadrant	Sign of $x$	Sign of $y$	Example
I (top right)	$+$	$+$	$(3, 2)$
II (top left)	$-$	$+$	$(- 2, 4)$
III (bottom left)	$-$	$-$	$(- 4, - 1)$
IV (bottom right)	$+$	$-$	$(2, - 3)$

A point with $y = 0$ lies on the $x$ -axis, and a point with $x = 0$ lies on the $y$ -axis. Points on an axis belong to no quadrant.

Important

The order in an ordered pair matters: $(2, 5)$ and $(5, 2)$ are different points. Always across first, then up or down.

Worked example

Plot the points $A (3, 2)$ , $B (- 2, 4)$ , $C (- 4, - 1)$ , $D (2, - 3)$ , $E (0, 3)$ and $F (- 3, 0)$ , and say where each one lies

$A (3, 2)$ : 3 right, 2 up, so quadrant I.
$B (- 2, 4)$ : 2 left, 4 up, so quadrant II.
$C (- 4, - 1)$ : 4 left, 1 down, so quadrant III.
$D (2, - 3)$ : 2 right, 3 down, so quadrant IV.
$E (0, 3)$ : no move across, 3 up, so it is on the $y$ -axis.
$F (- 3, 0)$ : 3 left, no move up, so it is on the $x$ -axis.

Worked example

Three corners of rectangle $P Q R S$ are $P (- 2, 1)$ , $Q (3, 1)$ and $R (3, 4)$ . Find $S$ , and the area of the rectangle

$P$ and $Q$ have the same $y$ -coordinate, so $P Q$ is horizontal. Its length is $3 - (- 2) = 5$ units.
$Q$ and $R$ have the same $x$ -coordinate, so $Q R$ is vertical. Its length is $4 - 1 = 3$ units.
$S$ must be straight above $P$ (same $x$ as $P$ ) and level with $R$ (same $y$ as $R$ ). So $S (- 2, 4)$ .
Area $= 5 \times 3 = 15$ square units.

Warning

Swapping the coordinates. A common slip is to plot $(2, - 3)$ by going 2 up first. Say it aloud as you plot: "across 2, down 3". The $x$ -coordinate always comes first, just as a corridor comes before the stairs.

Tip

Memory hook: "Along the corridor, then up the stairs." You walk along ( $x$ ) before you climb ( $y$ ). Alphabetical order works too: $x$ comes before $y$ .

Math Antics (mathantics): "Algebra Basics: Graphing On The Coordinate Plane". Watch how the two number lines are placed to make the plane, and why the order of the pair matters.

✦

Practice

Work in your exercise book. Questions get harder as you go.

Write the coordinates of the origin.
Arrange in order, smallest first: $1.5,\ -2,\ -0.5,\ 3,\ -3.5$ .
Where does each point lie: $(- 5, 2)$ , $(4, - 1)$ , $(- 3, - 7)$ , $(6, 0)$ ?
A point is 4 units to the left of the $y$ -axis and 3 units below the $x$ -axis. Write its coordinates.
Plot $P (1, 4)$ , $Q (1, - 2)$ and $R (- 3, - 2)$ and join them. What shape is $P Q R$ , and at which corner is its right angle?
Three corners of a square are $(- 1, - 1)$ , $(3, - 1)$ and $(3, 3)$ . Find the fourth corner and the area of the square.
Kalisa says $(2, - 5)$ and $(- 5, 2)$ are the same point because they use the same two numbers. Is he right? Explain.

Show answersHide answers

$(0, 0)$
$-3.5,\ -2,\ -0.5,\ 1.5,\ 3$
Quadrant II; quadrant IV; quadrant III; on the $x$ -axis (no quadrant).
$(- 4, - 3)$
A right-angled triangle. $P Q$ is vertical and $Q R$ is horizontal, so the right angle is at $Q$ .
$(- 1, 3)$ . Each side is 4 units, so the area is $4^{2} = 16$ square units.
No. $(2, - 5)$ is in quadrant IV and $(- 5, 2)$ is in quadrant II. An ordered pair is read in order: the first number is always $x$ .

Sources and credits

4.1: Use the Rectangular Coordinate System, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Algebra Basics: Graphing On The Coordinate Plane, mathantics, embedded from YouTube.

The worked examples, diagrams, practice items and quiz questions were written by Ganzaa.

11. The number line

22. Two number lines make a plane

33. Drawing and labelling axes

44. Signs in the four quadrants

1. The number line

2. Two number lines make a plane

3. Drawing and labelling axes

4. Signs in the four quadrants