Modulus Functions (Cambridge O Level Additional Maths)

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Sketching Modulus Graphs

What is a modulus function?

  • The modulus function makes any ‘input’ positive
    • This is sometimes called the absolute value (of the input)
    • The modulus function is indicated by a pair of vertical lines being written around the input
      • Similar to how brackets are used
      • e.g.  vertical line 7 vertical line equals 7 comma space space vertical line minus 7 vertical line equals 7

What types of modulus graphs will I need to sketch?

  • Modulus graphs required will be of linear, or quadratic form
    • Linear form will be vertical line a x plus b vertical line
    • Quadratic form will be vertical line a x squared plus b x plus c vertical line
  • Often, there will be two graphs to sketch as this helps with solving equations involving modulus functions
    • In linear form, equations could be of the form vertical line a x plus b vertical line equals vertical line c x plus d vertical line
      • One side of the equation may not involve the modulus
      • One side may have a constant term only (i.e. b equals 0 and/or d equals 0
    • In quadratic form, equations could be of the form vertical line a x squared plus b x plus c vertical line equals d
  • In both cases, graphs of the left hand side and right hand side drawn on the same diagram will reveal the number of intersections of the graphs (and so the number of solutions to the equation) 

Modulus equations and using their graphs to find the number of intersections

How do I sketch the graph of the modulus of a function: y = |f(x)|?

STEP 1
Pencil in the graph of y = f(x)

STEP 2
Reflect anything below the x-axis, in the x-axis, to get y = |f(x)| Modulus Functions - Sketching Graphs Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

  • Note in particular that the y-axis intercept, if negative on the graph of y equals straight f open parentheses x close parentheses will be positive on the graph of y equals vertical line straight f open parentheses x close parentheses vertical line
  • At the x-axis intercepts, the graph will have a sharp logical or-shape
    • this is not a smooth curve like with a turning point
    • if the graph is of the form y equals negative vertical line straight f open parentheses x close parentheses vertical line then the graph would be wholly negative and the x-axis intercepts would have a sharp logical and-shape

Worked example

sketching modulus graphs worked example

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Solving Modulus Equations

Why are graphs needed to solve modulus equations?

  • Sketching the graphs of two modulus function(s) on the same diagram quickly reveals
    • the number of solutions there are to the equation
    • which parts of the graph (equation) - either the 'normal' part or the 'reflected' part - will be needed to solve equations
  • For example, two non-parallel linear graphs would intersect
    • the solution to the linear functions being equal to each other would have one solution 
    • if a modulus is involved there could be more than one intersection/solution

number of intersections of two linear graphs

How do I solve modulus equations?

STEP 1
Sketch the graphs including any modulus (reflected) parts

STEP 2
Locate the graph intersections

STEP 3

Determine which part of each graph ('normal' or 'reflected' part) is needed to solve the equation
Solve the appropriate equation(s)
worked example - solving two linear modulus functions equal to each other

How do I solve modulus inequalities?

  • The process is very similar to that as solving equations - with the graph sketching being essential

STEP 1
Sketch the graph(s) including any modulus (reflected) parts

STEP 2
Locate the graph intersections (or x-axis intercepts if zero on one side)

STEP 3
Determine which part(s) of the graph(s) satisfy the inequality (highlight any on the graph)
Find the intersections (by solving equation(s))

STEP 4
Write the final answer(s) down, being careful with the use of <, >, ≤ and/or ≥

Steps to solve a more complicated modulus equation

Examiner Tip

  • Sketching the graphs is important as solving algebraically can lead to invalid solutions
    • For example, x = 1 is a solution to x minus 4 equals 2 x minus 5
    • but x equals 1 is not a solution to vertical line x minus 4 vertical line equals 2 x minus 5
      • (substitute x = 1 into both sides and see why it does not work)

Worked example

Worked example - solving modulus functions equation and inequality part aWorked example - solving modulus functions equation and inequality part b

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.