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123In graphs

Theory

A function f with function description y = f ( x ) is

  • increasing if the function values increase as x increases;

  • decreasing if the function values decrease as x increases.

Furthermore, the function has

  • a maximum when it goes from increasing to decreasing and the graph is continuous;

  • a minimum when it goes from decreasing to increasing and the graph is continuous.

The maximum and the minimum values are called the extreme values of the function.

The behaviour of a function can be classified more precisely:

  • increasing (decreasing) constantly when the rate at which the function increases (decreases) is constant

  • increasingly increasing (decreasing) when the slope of the curve becomes steeper;

  • decreasingly increasing (decreasing) when the slope of the curve becomes less steep.

To indicate where a function is decreasing/increasing you use intervals.

An interval is a part of the number line. You denote an interval by writing down the boundary values (separated by a comma, the smallest value first) between two parentheses. Two things are important to remember:

  • the shape of the parentheses determines whether the boundary value is part of the interval or not;

  • use an arrow for intervals that do not have a boundary at one end.

Here are some examples of intervals and the corresponding part of the number line.

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