Graphs and Coordinate Geometry (Mathematics Curriculum)

What Is a Graph?

A graph is a diagram that allows us to draw points, lines, curves and other shapes.

A graph uses a pair of axes at right angles to each other: one horizontal and one vertical.

We measure how far across each point on the graph is using the horizontal x-axis. We measure how far up each point on the graph is using the vertical y-axis.

Dictionary Definition

The Merriam-Webster dictionary defines a graph as "a diagram (such as a series of one or more points, lines, line segments, curves, or areas)."

A graph is shown below:

What Is Coordinate Geometry?

Coordinates are used to describe the position of a point on the graph.

  • The x-coordinate measures how far across the point is (in the horizontal direction) along the x-axis.

    In general, if the point is x units along the x-axis, its x-coordinate is x.

  • The y-coordinate measures how far up the point is (in the vertical direction) up the y-axis.

    In general, if the point is y units up the y-axis, its y-coordinate is y.

Each point can be written as a pair of coordinates:

The x-coordinate (x) and the y-coordinate (y) are then written in brackets, separated by a comma. The x-coordinate is on the left, the y-coordinate is on the right.

The coordinates can be used to describe points on lines, curves and shapes. This is coordinate geometry.

Here are some examples from coordinate geometry. We might be interested in the distance between two points. We might want be interested in how a shape is moved. Coordinate geometry allows us to do this.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning.

A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Graphs

A graph is a diagram that allows us to draw points, lines, curves and other shapes.

A graph uses a pair of axes at right angles to each other: one horizontal and one vertical.

In this mini-curriculum, you will learn about the parts of a graph.

Parts of a Graph

The parts of a graph are the two axes (the horizontal x-axis and the vertical y-axis) that meet at right angles at the origin.

X-Axis

The x-axis is the horizontal axis on a graph.

Y-Axis

The y-axis is the vertical axis on a graph.

Origin

The origin is where the x-axis and y-axis meet on a graph.

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Cartesian Coordinates

Cartesian coordinates are used to describe the position of a point on a graph.

In this mini-curriculum, you will learn about Cartesian coordinates and how they are used to describe and draw points.

Cartesian Coordinates

Cartesian coordinates are used to describe the position of a point on a graph.

X-Coordinate

The x-coordinate is the first number in the pair of numbers used to describe Cartesian coordinates. It tells you how far across the horizontal x-axis a point is on a graph.

Y-Coordinate

The y-coordinate is the second number in the pair of numbers used to describe Cartesian coordinates. It tells you how far up (or down) the vertical y-axis a point is on a graph.

Draw a Point from Cartesian Coordinates

To draw a point from Cartesian coordinates, use the x-coordinate to measure how far across the x-axis the point is and use the y-coordinate to measure how up (or down) the y-axis the point is.

Read Off a Point from Cartesian Coordinates

To read off a point from Cartesian coordinates, measure how far across the x-axis the point is to find the x-coordinate and measure how up (or down) the y-axis the point is to find the y-coordinate.

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Polar Coordinates

Polar coordinates are another coordinate system. They are used to describe the position of a point on a graph.

In this mini-curriculum, you will learn about polar coordinates.

Polar Coordinates

Polar coordinates are used to describe the position of a point on a graph.

Polar coordinates work by measuring how far the point is from a reference point (called the pole) and what angle it is from a reference direction (called the polar axis).

Radial Coordinate

The radial coordinate is the first number in the pair of numbers used to describe polar coordinates. It tells you how far a point is from the pole.

Angular Coordinate

The angular coordinate is the second number in the pair of numbers used to describe polar coordinates. It tells you what angle the point is (in the counter-clockwise direction) from the polar axis.

Draw a Point from Polar Coordinates

To draw a point from polar coordinates, use the radial coordinate to measure how far the point is from the pole and use the angular coordinate to find what angle the point is (in the counter-clockwise direction) from the polar axis.

Read Off a Point from Polar Coordinates

To read off a point from polar coordinates, measure how far across the point is from the pole to find the radial coordinate and find what angle the point is (in the counter-clockwise direction) from the polar axis to find the angular coordinate.

Convert Cartesian to Polar Coordinates

Cartesian coordinates can be converted to polar coordinates.

Convert Polar to Cartesian Coordinates

Polar coordinates can be converted to Cartesian coordinates.

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Coordinate Geometry of Points and Lines

Coordinate geometry can be used for points and lines that join any two points.

In this mini-curriculum, you will learn about finding the distance and slope between two points and finding the midpoint between two points.

Distance Between Points

The distance between two points, where (x1, y1) and (x2, y2) are the Cartesian coordinates of the points, is:

Slope Between Points

The slope between two points, where (x1, y1) and (x2, y2) are the Cartesian coordinates of the points, is:

Find the Midpoint

The midpoint of two points, where (x1, y1) and (x2, y2) are the Cartesian coordinates of the points, is:

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Transformations in Coordinate Geometry

A transformation is a change of a shape. It can change a shape's position, orientation and its size.

Cartesian coordinates can be used to transform a shape.

In this mini-curriculum, you will learn about how to use coordinate geometry to move and flip shapes.

Translate a Shape

A translation moves a shape.

To translate a shape using coordinate geometry, change the x-coordinate of each point by how far you move the shape horizontally and change the y-coordinate of each point by how far you move the shape vertically.

Reflect a Shape in X-Axis

A reflection is a flip of a shape about a line (called the line of reflection).

To reflect a shape in the x-axis using coordinate geometry, the x-coordinate stays the same, but the y-coordinate changes sign (becomes negative if it is positive and vice versa).

Reflect a Shape in Y-Axis

A reflection is a flip of a shape about a line (called the line of reflection).

To reflect a shape in the y-axis using coordinate geometry, the y-coordinate stays the same, but the x-coordinate changes sign (becomes negative if it is positive and vice versa).

Reflect a Shape in y = x

A reflection is a flip of a shape about a line (called the line of reflection).

To reflect a shape in the line y = x using coordinate geometry, the x-coordinate becomes the y-coordinate and the y-coordinate becomes the x-coordinate.

Reflect a Shape in y = −x

A reflection is a flip of a shape about a line (called the line of reflection).

To reflect a shape in the line y = −x using coordinate geometry

  • both coordinates change sign (the coordinate becomes negative if it is positive and vice versa)

  • the x-coordinate becomes the y-coordinate and the y-coordinate becomes the x-coordinate

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Equation of a Circle

A circle can be represented with an equation.

The Cartesian coordinates of the points on the circle satisfy this equation.

In this mini-curriculum, you will learn about the equation of a circle.

Basic Equation of Circle

The basic equation of a circle where (x, y) are the Cartesian coordinates of the points on the circle and r is the radius is:

Equation of Circle

The equation of a circle where (x, y) are the Cartesian coordinates of the points on the circle, (a, b) are the Cartesian coordinates of the center of the circle and r is the radius is:

Find Center and Radius from Equation of Circle

The center and radius of a circle can be found from the equation of a circle.

Equation of Circle in General Form

The equation of a circle in general form where (x, y) are the Cartesian coordinates of the points on the circle is:

Convert General to Standard Form

A circle in general form can be converted to standard form.

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