S-Cool Revision Summary

S-Cool Revision Summary

The basics

A function can be one-to-one (one x value gives one y-value), or many-to-one (more than one x-value can give the same y-value)

Domain = values x can take.

Range = values y can take.

To sketch functions find:

1.

Where the graph crosses the y-axis.

The graph crosses the y-axis when x = 0. (i.e. at the constant).

2.

Where the graph crosses the x-axis.

To find the roots (where the graph crosses the x-axis), 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 y-axis.

If f(-x) = f(x), then the graph is an even function (symmetrical about the y-axis).

-f(x) is a reflection of f(x) in the x-axis.

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 y-axis.

f(ax) is a stretch scale factor in the x-axis.

Inverse Function

The inverse function is found by reflecting the function in the line y = x, and can be calculated by:

  1. Write the equation as y = f(x).

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

  3. Rearrange the formula into a new y = f(x). This is the inverse function.

Remember: only one-to-one functions produce inverse functions, so remember to limit the domain.