Introduction

A graph is a form of representing and can be described as an equation (eg. y=4x+2) or as a function (eg. f(x)=4x+2) The 2 examples given would describe the same graph.
To the right are examples of the following graphs (made using desmos as for graphs throughout this page)...
$$ y = x $$
$$ y = x^{2} $$
$$ y = x^{3} $$
$$ y = x^{4} $$
Background Knowledge
Graph plotting and sketching
Functions of Graphs and Transformations of Functions
Algebraic Manipulation and solving Algebraic Equations
Trigonometry

Linear, Quadratic, Cubic and Quartic Graphs

When asked to plot any graph, work out the values of y for various values of x then plot each of the coordinates. Then, for non-linear graphs, join the points with the appropriate curves. (Draw a straight line for linear graphs with a ruler). Below are examples of each type of graph.
When asked to sketch a graph, work out the coordinates of key points such as the y-intercept, where it crosses the x axis and sometimes the turning point (typically acquired by completing the square for quadratics).
The y-intercept is where the line crosses the y axis. The y axis is the same as the line x=0 so the value of x has to be 0 on this line. Therefore, to find the y axis, substitute 0 for x.The x axis is the same as the line y=0 so the value of y on this line must be 0. In this case, you are solving for x.
Example:
$$ f(x) = x^{3}+2x^{2}-x-2 $$
This factorises to (x-1)(x+1)(x+2)

1) The y-intercept
  • For this, x=0
  • Substitute into f(x)
  • f(0) = -2
  • y-intercept is (0,-2)
2) Points on the x axis
  • For this, y = 0
  • This means f(x) = 0
  • Solve for x
  • (x-1)(x+1)(x+2) = 0
  • x = 1, x= -1, x= -2
  • Coordinates: (1,0), (-1,0), (-2,0)
​3) Sketch is shown to left, joining the points with a smoothly drawn cubic curve.

Exponential Graphs

An exponential function (graph) is one where the variable (x) is a power. For example...
$$ y = 2^{x} $$
​​$$ y = -3^{6-3x}+1 $$
​​$$ f(x) = -3^{6-3x}+1 $$
  • Important Points
  • Transforming Functions
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The power will be equal to 0 when x is 2. In this instance, the y coordinate would then be 0 as -(1)+1 = 0
Due to the negative coefficient of x, the graph decreases exponentially and so the larger x gets, the smaller 3^{6-3x} gets and as that approaches 0, the closer f(x) gets to 1
In terms of functions, we can consider this graph a transformation of our graph from earlier.
  • ​However, some values change due to the change in power
  • ​The function/graph is reflected in the y axis
  • The function/graph is translated such that it rises by 1 
$$ f(x) = 2^{x} $$
​For any exponential function, if the power is 0, the value is 1 as any number to the power of 0 is equal to 1. This means, when x=0 on this graph (at the y axis), y = 1
In this graph, as x (the power) increases, the value of y increases at an increasing rate (exponentially).
As x decreases, the value of y also decreases and the smaller the value of x, the closer y is to 0. y approaches 0 but does not reach 0 and so gets smaller asymptotically.
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You can test these theories by imputing different values of x
The graph is shown to the left.

Trigonometric Functions

The Unit Circle

Sine, Cosine and Tangent Graphs
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