Increasing and Decreasing Functions for IGCSE Maths
Determining where a function is increasing or decreasing. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding increasing and d
What You Need to Know
Determining where a function is increasing or decreasing. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding increasing and decreasing functions is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Increasing and Decreasing Functions
A function y = f(x) is increasing on an interval when dy/dx > 0 (the gradient is positive), and decreasing when dy/dx < 0. Finding these intervals requires solving the inequality dy/dx > 0 or dy/dx < 0 — often by solving the quadratic dy/dx = 0 first to find the boundary points, then testing regions. IGCSE Extended Paper 4 asks students to state or find the range of x values for which a function is increasing or decreasing.
Step-by-Step Method
- 1
Differentiate
Find dy/dx.
- 2
Find the zeros of dy/dx
Solve dy/dx = 0 to find the x-values where the gradient is zero (the boundaries).
- 3
Determine the sign of dy/dx in each region
Test a value in each region between and beyond the roots. Positive → increasing; negative → decreasing.
- 4
Write the intervals
State the x-ranges using inequalities: 'increasing for x < a or x > b' or 'decreasing for a < x < b'.
- 5
Relate to the graph
Sketch the curve or gradient function to visualise the increasing/decreasing behaviour.
Worked Example
Question
y = x³ − 6x² + 9x − 2. Find the values of x for which y is a decreasing function.
Solution
Step 1: dy/dx = 3x² − 12x + 9 = 3(x² − 4x + 3) Step 2: dy/dx = 0 → 3(x−1)(x−3) = 0 → x=1 or x=3 Step 3: Test regions: - x=0: dy/dx = 3(0−1)(0−3) = 3(−1)(−3) = 9 > 0 → increasing - x=2: dy/dx = 3(2−1)(2−3) = 3(1)(−1) = −3 < 0 → decreasing - x=4: dy/dx = 3(3)(1) = 9 > 0 → increasing Answer: Decreasing for 1 < x < 3
Exam Tips for Increasing and Decreasing Functions
- Increasing ↔ positive gradient (dy/dx > 0); decreasing ↔ negative gradient (dy/dx < 0).
- Find the roots of dy/dx = 0 first — these divide the x-axis into regions to test.
- You can use a sign diagram or number line to organise the sign of dy/dx in each region.
- Verify by connecting to stationary points: a function decreases between a maximum and the next minimum.
Practice Questions
Q1: y = 2x³ − 3x² − 12x + 5. Find the interval where y is increasing.
Show hint
dy/dx = 6x²−6x−12 = 6(x−2)(x+1) = 0. Increasing when dy/dx>0: x<−1 or x>2.
Q2: y = −x² + 6x. For what values of x is y increasing?
Show hint
dy/dx = −2x+6>0 → 6>2x → x<3. Increasing for x<3.
Q3: f(x) = x⁴ − 4x² + 3. Find all x-values where f is decreasing.
Show hint
f'(x)=4x³−8x=4x(x²−2)=0. Roots: x=0,±√2. Test signs in each region.
Frequently Asked Questions
What is increasing and decreasing functions in IGCSE Maths?
Determining where a function is increasing or decreasing.
Is increasing and decreasing functions in the Core or Extended syllabus?
Increasing and Decreasing Functions is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise increasing and decreasing functions effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on increasing and decreasing functions rather than trying to cover everything at once.
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