Modulus Transformations

STEP 1: Keep the parts of the graph of y = f(x) that are on or above the x-axis
STEP 2: Any parts of the graph below the x-axis get reflected in the x-axis

The transformations outside the function follow the same order as the order of operations
- $y = |a f (x) + b|$
  - Deal with the a then the b then the modulus
- $y = a |f (x)| + b$
  - Deal with the modulus then the a then the b
The transformations inside the function are in the reverse order to the order of operations
- $y = f (|a x + b|)$
  - Deal with the modulus then the b then the a
- $y = f (a |x| + b)$
  - Deal with the b then the a then the modulus

When sketching one of these transformations in an exam question make sure that the graphs do not look smooth at the points where the original graph have been reflected
- For $y = |f (x)|$ the graph should look "sharp" at the points where it has been reflected on the x-axis
- For $y = f (|x|)$ the graph should look "sharp" at the point where it has been reflected on the y-axis

The diagram below shows the graph of $y = f (x)$ .

2-9-1-we-diagram

(a) Sketch the graph of $y = |f (x)|$ .

2-9-1-ib-aa-hl-modulus-trans-a-we-solution

(b) Sketch the graph of $y = f (|x|)$ .

2-9-1-ib-aa-hl-modulus-trans-b-we-solution

Modulus Transformations (DP IB Maths: AA HL)