IntroductionA 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 sketch a graph, work out the coordinates of key points such as the yintercept, where it crosses the x axis and sometimes the turning point (typically acquired by completing the square for quadratics).
The yintercept 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: $$ f(x) = x^{3}+2x^{2}x2 $$ This factorises to (x1)(x+1)(x+2) 1) The yintercept

Exponential Graphs
An exponential function (graph) is one where the variable (x) is a power. For example...
$$ y = 2^{x} $$
$$ y = 3^{63x}+1 $$ $$ f(x) = 3^{63x}+1 $$
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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^{63x} gets and as that approaches 0, the closer f(x) gets to 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. 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
Sine, Cosine and Tangent Graphs