SCool Revision Summary
SCool Revision Summary
The basics
A function can be onetoone (one x value gives one yvalue), or manytoone (more than one xvalue can give the same yvalue)
Domain = values x can take.
Range = values y can take.
To sketch functions find:
1. 
Where the graph crosses the yaxis. 
The graph crosses the yaxis when x = 0. (i.e. at the constant). 
2. 
Where the graph crosses the xaxis. 
To find the roots (where the graph crosses the xaxis), we solve the equation y = 0. 
3. 
Where the stationary points are. 
The stationary points occur when the gradient is 0. (i.e. differentiate.) 
4. 
Whether there are any discontinuities 
A discontinuity occurs when the graph is undefined for a certain value of x. This occurs when x appears in the denominator of a fraction (you can't divide by zero). 
5. 
What happens as c ® ± ¥ 
When x becomes a large positive or negative number the graph will tend towards a certain value. 
Transformations
f(x) + k is a translation of f(x) by the vector
f(x + a) is a translation of f(x) by the vector
Therefore:
f(x) = f(x  a) + b is a translation of f(x)
by the vector
f(x) is a reflection of f(x) in the yaxis.
If f(x) = f(x), then the graph is an even function (symmetrical about the yaxis).
f(x) is a reflection of f(x) in the xaxis.
If f(x) =  f(x), then the graph is an odd function (rotational symmetry about the origin).
af(x) is a stretch scale factor a in the yaxis.
f(ax) is a stretch scale factor in the xaxis.
Inverse Function
The inverse function is found by reflecting the function in the line y = x, and can be calculated by:

Write the equation as y = f(x).

Swap the letters x and y. (This is the same as reflecting in the line y = x.)

Rearrange the formula into a new y = f(x). This is the inverse function.
Remember: only onetoone functions produce inverse functions, so remember to limit the domain.