# Year 10+ Coordinate Geometry

### Chapters

### Finding the Graph for an Equation

# Finding the Graph for an Equation

The graph for an equation is the set of points at which the equation is satisfied (that is, the \(x\) and \(y\)-values that work!).

Let's look at a few examples.

## Example 1

To find the graph of the equation \(y= 2x + 1\), we start by plotting some points at which the equation is satisfied:

\(x\) | \(y = 2x + 1\) |
---|---|

\(-2\) | \(-3\) |

\(-1\) | \(-1\) |

\(0\) | \(1\) |

\(1\) | \(3\) |

\(2\) | \(5\) |

In this example, joining all these points will give the graph of the function, which is the set of **all points** satisfying the equation. Of course, we can only show a finite
section of the graph. This graph will go on forever in both directions. Here's part of the graph of the equation \(y = 2x + 1\):

## Example 2

To find the graph of the equation \(y= x^2 - 3\), we also start by plotting some points at which the equation is satisfied:

\(x\) | \(y = x^2 - 3\) |
---|---|

\(-2\) | \(1\) |

\(-1\) | \(-2\) |

\(0\) | \(-3\) |

\(1\) | \(-2\) |

\(2\) | \(1\) |

In this example, these points and our knowledge of the graph of equations like this give us a good idea of the graph of the function, which is the set of **all points** satisfying the equation. Of course, we can only show a finite
section of the graph. It is best to plot as many points as possible. This graph will extend forever upwards. Here's part of the graph of the equation \(y = x^2 - 3\):

Now let's look at an example in which plotting just a few points can be misleading.

## Example 3

To find the graph of the equation \(y= x^3 - 2x + 2\), we also start by plotting some points at which the equation is satisfied. If we start with just these points

\(x\) | \(y = x^3 - 2x + 2\) |
---|---|

\(-2\) | \(-2\) |

\(0\) | \(2\) |

\(2\) | \(6\) |

If we'd plotted a few more points satisfying the equation before we tried to draw the graph, we'd have realised that the graph actually looks more like this:

## More Suggestions for Graphing Equations

If you get really stuck, you can use one of the many free on-line graphing calculators that are available. I like Desmos and GeoGebra, but there are quite a few of them. It's good to play around with them to get an idea of what the graphs of various types of equations look like.

It's also a good idea to learn to recognise the equations of some common types of graphs like straight lines, parabolas and circles.

A really good graph of an equation should include the following features:

- \(x\) and \(y\)-intercepts.
- Maximum values (peaks)
- Minimum values (valleys)
- Any areas where the graph plateaus (is flat)
- Asymptotes (lines that the graph gets as close as you like to, but never crosses)
- Any gaps or jumps
- A few plotted points
- Where is the graph increasing (going uphill) or decreasing (going downhill)?
- What happens to the graph for very large and very small values of \(x\)?
- Any other special features of the equation

In Years 11 and 12, you'll learn about Calculus. Calculus provides you with a wonderful tool kit for finding the important features of equations, and so will help you to draw their graphs.

### Description

A **coordinate geometry** is a branch of **geometry** where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as **coordinates**. In this tutorial series, you will learn about vast range of topics such as Cartesian Coordinates, Midpoint of a Line Segment etc

### Audience

year 10 or higher, several chapters suitable for Year 8+ students.

### Learning Objectives

Explore topics related to Coordinates Geometry

Author: Subject Coach

Added on: 27th Sep 2018

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