CIE A Level Maths: Pure 3复习笔记1.1.1 Modulus Functions - Sketching Graphs

Modulus functions

• The modulus function makes any ‘input’ positive
• |x| = x   if x ≥ 0   |f(x)| = f(x)   if f(x) ≥ 0
• |x| = -x if x < 0   |f(x)| = -f(x)   if f(x) < 0
  • For example: |5| = 5 and |-5| = 5
• Sometimes called absolute value

How do I sketch the graph of the modulus function: y = a |­x + p| + q?

• The graph will look like a “ꓦ” if a > 0 or a “ꓥ” if a < 0
• There will be a vertex at the point (-p, q)
• There could be 0, 1 or 2 roots
  • This depends on the location of the vertex and the orientation of the graph (ꓦ or ꓥ)
• Compare this to the completed square form of a quadratic a(x + p)² + q

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)|

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

STEP 1          Sketch the graph of y = f(x) only for x ≥ 0

STEP 2          Reflect this in the y-axis

What is the difference between y = |f(x)| and y = f(|x|)?

• There is a difference between y = |f(x)| and y = f(|x|)
• The graph of y = |f(x)| never goes below the x-axis
  • It does not have to have any lines of symmetry
• The graph of y = f(|x|) is always symmetrical about the y-axis
  • It can go below the y-axis

• For the exam you will only be asked to do this when f(x) is linear
  • Your graphs will all look like a “ꓦ” or a “ꓥ”
  • You can also think of these graphs as transformations of the graph y = |x|