Modulus Functions (Cambridge International IGCSE Further Maths)

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. $\left|7\right| =7 , \left|−7\right| =7$

What types of modulus graphs will I need to sketch?

• Modulus graphs required will be of linear, or quadratic form

  • Linear form will be $\left|a x + b\right|$
  • Quadratic form will be $\left|a x²+b x+c\right|$

• 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 $\left|a x+b\right| = \left|c x+d\right|$

    • One side of the equation may not involve the modulus
    • One side may have a constant term only (i.e. $b=0$ and/or $d=0$

  • In quadratic form, equations could be of the form $\left|a x²+b x+c\right| =d$

• 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)

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)|

• Note in particular that the $y$-axis intercept, if negative on the graph of $y= \mathrm{f} (x)$ will be positive on the graph of $y= \left|\mathrm{f} (x)\right|$

• 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 $y=− \left|\mathrm{f} (x)\right|$ then the graph would be wholly negative and the $x$-axis intercepts would have a sharp logical and-shape

Worked Example

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

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)

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 ≥

Exam Hint

• Sketching the graphs is important as solving algebraically can lead to invalid solutions

  • For example, x = 1 is a solution to $x-4=2x-5$

  • but $x=1$ is not a solution to $\left|x−4\right| =2x−5$

    • (substitute x = 1 into both sides and see why it does not work)

Worked Example