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

2. 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

3. 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

4. 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.

5. The graphs of the increasing functions and are

6. 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.

7. 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.

8. 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.