Increasing and decreasing functions Revision Notes for AQA GCSE Further Maths

Increasing and decreasing functions

What are increasing and decreasing functions?

Understanding whether a function is increasing or decreasing is fundamental in calculus. When we say a function is increasing, we mean that as the x-values get larger, the y-values also get larger. Conversely, a decreasing function means that as x-values increase, the y-values get smaller.

The key to determining this behaviour lies in examining the derivative of the function. The derivative tells us the rate of change of the function at any point, which directly indicates whether the function is climbing upward or falling downward.

Note

The derivative acts like a "slope detector" - when it's positive, the function is going uphill (increasing), and when it's negative, the function is going downhill (decreasing).

Mathematical conditions

Important

Fundamental Rules for Function Behavior

For any function $y = f(x)$ , we can determine its behaviour using these essential rules:

A function $y = f(x)$ is:

Increasing when the derivative $\frac{dy}{dx} > 0$ (positive derivative)
Decreasing when the derivative $\frac{dy}{dx} < 0$ (negative derivative)

This makes perfect sense when you think about it - a positive derivative means the function is rising, while a negative derivative means it's falling.

Some functions maintain the same behaviour across their entire domain. For instance, the linear function $y = 3 - 2x$ is always decreasing because its derivative $\frac{dy}{dx} = -2$ , which is always negative.

However, many functions change their behaviour over different parts of their domain. These functions may be increasing in some intervals and decreasing in others, which is where our derivative analysis becomes particularly valuable.

Worked example: Finding when a function is increasing

Example

Worked Example: Determining Increasing Intervals

Question: Work out the values of x for which the function $y = x^2 - 4x + 1$ is an increasing function.

Solution:

Step 1: Find the derivative First, we need to differentiate the function: $\frac{dy}{dx} = 2x - 4$

Step 2: Set up the inequality for increasing behaviour For the function to be increasing, we need: $\frac{dy}{dx} > 0$

Substituting our derivative: $2x - 4 > 0$

Step 3: Solve the inequality $2x - 4 > 0$ $2x > 4$ $x > 2$

Answer: The function $y = x^2 - 4x + 1$ is increasing when x > 2.

Key problem-solving steps

Note

Systematic Approach to Function Behaviour Problems

When tackling problems about increasing and decreasing functions, follow this systematic approach:

Differentiate the function to find $\frac{dy}{dx}$
Determine what you're looking for - increasing ( $\frac{dy}{dx} > 0$ ) or decreasing ( $\frac{dy}{dx} < 0$ )
Set up the appropriate inequality using your derivative
Solve the inequality to find the range of x-values
Express your answer clearly using interval notation or inequality statements

Common exam tips and techniques

Note

Watch out for these common traps:

Don't forget to differentiate correctly - check your basic differentiation rules
Remember that increasing means $\frac{dy}{dx} > 0$ , not $\frac{dy}{dx} ≥ 0$ (the equals sign makes a difference)
Be careful with inequality signs when multiplying or dividing by negative numbers

Note

Typical problem-solving approaches:

Always start by finding the derivative - this is your key tool
When solving inequalities, treat them like equations but remember to flip inequality signs when multiplying/dividing by negatives
Sketch a quick sign chart if you're dealing with more complex expressions to visualise where the derivative is positive or negative

Note

Exam-style tips:

Show all your working clearly, especially when solving inequalities
State your final answer in the form requested (e.g., " $x > 2$ " or "the function is increasing for $x > 2$ ")
Double-check your differentiation - this is where most errors occur

Remember!

Summary

Key Points to Remember:

A function is increasing when $\frac{dy}{dx} > 0$ (positive derivative means going upward)
A function is decreasing when $\frac{dy}{dx} < 0$ (negative derivative means going downward)
The process is always: differentiate → set up inequality → solve → state answer clearly
Some functions are always increasing or decreasing, while others change behaviour in different intervals
Pay careful attention to inequality signs and show all working steps in exam questions

Increasing and decreasing functions (AQA GCSE Further Maths): Revision Notes