“Teach A Level Maths” Vol. 1: AS Core Modules 12a: Increasing and Decreasing Functions © Christine Crisp

An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x

e.g.1 Show that is an increasing function Solution: is the sum of a positive number ( 3 )  a perfect square ( which is positive or zero for all values of x, and a positive number ( 4 ) for all values of x so, is an increasing function

e.g.2 Show that is an increasing function. Solution: To show that is never negative ( in spite of the negative term ), we need to complete the square. for all values of x Since a square is always greater than or equal to zero, is an increasing function.

The graphs of the increasing functions and are

Exercises 1. Show that is a decreasing function and sketch its graph. 2. Show that is an increasing function and sketch its graph. Solutions are on the next 2 slides.

Solutions 1. Show that is a decreasing function and sketch its graph. Solution: . This is the product of a square which is always and a negative number, so for all x. Hence is a decreasing function.

Solutions 2. Show that is an increasing function and sketch its graph. Solution: . Completing the square: which is the sum of a square which is and a positive number. Hence y is an increasing function.