Functions: Coordinate System

Learn the Cartesian coordinate system: x and y axes, quadrants, the origin, and how to plot and locate points with interactive math practice.

✔️Solved: 0

Learning Guide: The Coordinate System

The Axes & The Origin

The grid is built from two intersecting axes:

x-axis: The horizontal axis. Its values increase as you move right (positive direction) and decrease as you move left (negative direction).
y-axis: The vertical axis. Its values increase as you move upwards (positive direction) and decrease as you move downwards (negative direction).
Intersection: The two axes cross each other at a 90° angle (they are perpendicular). The point where they cross is called the origin, represented by coordinates $(0, 0)$ .

Figure 1: The coordinate plane showing the x-axis, y-axis, origin, the four quadrants, and a plotted point at (3, 2).

Writing points as Ordered Pairs

Any point on the coordinate plane is represented as an ordered pair written in the format:

(x, y)

The first number ( $x$ ) is the x-coordinate (abscissa). It tells you how many units to move horizontally from the origin (right if positive, left if negative).
The second number ( $y$ ) is the y-coordinate (ordinate). It tells you how many units to move vertically from the origin (up if positive, down if negative).

Coordinates of a Point

(x, y)

x: horizontal position
y: vertical position

The Four Quadrants (Regions)

The two crossing axes divide the coordinate system into four regions called quadrants, numbered counter-clockwise starting from the top-right:

First Quadrant (I): Top-right region. Both $x$ and $y$ are positive ( $x > 0, y > 0$ ). Example: $(2, 3)$ .
Second Quadrant (II): Top-left region. $x$ is negative and $y$ is positive ( $x < 0, y > 0$ ). Example: $(-4, 1)$ .
Third Quadrant (III): Bottom-left region. Both $x$ and $y$ are negative ( $x < 0, y < 0$ ). Example: $(-2, -3)$ .
Fourth Quadrant (IV): Bottom-right region. $x$ is positive and $y$ is negative ( $x > 0, y < 0$ ). Example: $(3, -5)$ .

Note: Points that lie directly on the x-axis or y-axis (like

(0, 4)

(-3, 0)

) or at the origin

(0, 0)

are not located in any of the four quadrants.

Midpoint of a Segment

The midpoint

M

of a segment connecting points

A(x_1, y_1)

and

B(x_2, y_2)

is the point that lies exactly halfway between them. The coordinates of the midpoint are given by the formula card below:

Midpoint coordinates of a segment

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

A: (x₁, y₁), B: (x₂, y₂)

Mathematical Derivation & Example:
To understand why this formula works, we can project the segment

AB

and its midpoint

M

onto the coordinate axes. This splits both the horizontal and vertical intervals into two equal parts:

1. Horizontal Projection ( $x$ -axis):
The horizontal interval spans from

x_1

x_2

. On our graph, this goes from

x_1 = -3

x_2 = 5

(a total distance of

8

units). The midpoint

M

projects to

x_M = 1

. Since the distance from

x_1

x_M

must equal the distance from

x_M

x_2

x_M - x_1 = x_2 - x_M

Using our graph coordinates:

1 - (-3) = 5 - 1 = 4\text{ units}

Solving algebraically for

x_M

2x_M = x_1 + x_2 \implies x_M = \frac{x_1 + x_2}{2}

Substituting the values yields:

x_M = \frac{-3 + 5}{2} = \frac{2}{2} = 1

2. Vertical Projection ( $y$ -axis):
The vertical interval spans from

y_1

y_2

. On our graph, this goes from

y_1 = -2

y_2 = 4

(a total distance of

6

units). The midpoint

M

projects to

y_M = 1

. Equalizing the distances:

y_M - y_1 = y_2 - y_M

Using our graph coordinates:

1 - (-2) = 4 - 1 = 3\text{ units}

Solving algebraically for

y_M

2y_M = y_1 + y_2 \implies y_M = \frac{y_1 + y_2}{2}

Substituting the values yields:

y_M = \frac{-2 + 4}{2} = \frac{2}{2} = 1

By combining these components, we find the exact coordinates of the midpoint:

M(1, 1)

The graph shows that projecting the segment onto the

x

and

y

axes divides the interval between coordinates into two equal parts:

x_M - x_1 = x_2 - x_M

and

y_M - y_1 = y_2 - y_M

Distance Between Two Points

The distance

d

between two points

A(x_1, y_1)

and

B(x_2, y_2)

is the length of the straight line segment connecting them. The distance formula is given by the formula card below:

Distance between two points

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

d: Distance, A(x₁, y₁), B(x₂, y₂)

Relation to the Pythagorean Theorem:
By projecting the segment

AB

onto the axes, we construct a right triangle

ACB

with a horizontal leg

AC

(length

|x_2 - x_1|

) and a vertical leg

CB

(length

|y_2 - y_1|

).
Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), the hypotenuse

d

satisfies:

d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \implies d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

For example, using the points

A(-3, -2)

and

B(5, 4)

on the graph:
- The horizontal distance is

x_2 - x_1 = 5 - (-3) = 8

.
- The vertical distance is

y_2 - y_1 = 4 - (-2) = 6

.
- Applying the theorem:

d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

The graph shows the right triangle

ACB

formed by projecting the segment

AB

onto the axes, demonstrating how the distance formula is derived directly from the Pythagorean theorem.

Learning Topics

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?

By mathematical convention, coordinates are always written in alphabetical order as

(x, y)

. This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.

What makes a relation a function?

A relation is a function if and only if each input value is associated with exactly one output value. If a single input has multiple different outputs, it is not a function.

How do you find where a function crosses the y-axis?

To find where a function crosses the

y

-axis, calculate

f(0)

by replacing

x

with

0

in the function formula. The resulting point on the graph will be

(0, f(0))

What happens if we input a value outside the domain?

If you input a value outside the domain, the function is undefined for that value. For example, in

f(x) = \frac{1}{x}

, inputting

x = 0

results in division by zero, which has no defined mathematical value.

How can you identify the domain of a function from its graph?

To find the domain from a graph, look at the graph's horizontal extent along the

x

-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).

What is the difference between domain and range?

The domain is the set of all valid input values (usually

x

) that you can feed into a function, while the range is the set of all output values (usually

y

) that the function produces as a result.

How can you identify the range of a function from its graph?

To find the range from a graph, look at the graph's vertical extent along the

y

-axis. Find the lowest and highest points of the graph, taking note of whether these endpoints are included (solid) or excluded (open circles).

What does it mean for a function to be continuous?

Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.

How do you find points where a function is not continuous?

Look for inputs that make the function undefined (like division by zero). For example,

f(x) = \frac{x^2 - 16}{x - 4}

is discontinuous at

x = 4

because you cannot divide by zero, creating a hole in the graph.

How do I determine whether a function is increasing or decreasing from its formula?

For linear functions

f(x) = mx + b

, the function increases if the slope

m > 0

and decreases if

m < 0

. For quadratic functions

f(x) = a(x-h)^2 + k

, check the sign of

a

: if

a > 0

, the parabola opens up, so it decreases for

x < h

and increases for

x > h

Why can't I use y-values to define intervals of increase or decrease?

Although we look at the vertical rise or fall (y-values) to determine *what* the graph does, the intervals must specify *where* it happens horizontally. By convention, intervals of increase/decrease partition the domain of the function, which is represented by the

x

-axis.