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 “?” if?a?> 0 or a “?” 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 (? or ?)
• 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 “?” or a “?”
  • You can also think of these graphs as?transformations?of the graph?y?= |x|