2.5PureStretch

Modulus Graphs

Two related sketches cause endless confusion: $y = |f(x)|$ and $y = f(|x|)$ . The modulus on the OUTSIDE reflects the negative part of the graph upwards; the modulus on the INSIDE makes the graph symmetric about the y-axis.

30 min Functions and Graphs

Edexcel A level Maths: 2.5 Sketching Modulus FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Sketch y = |f(x)| from y = f(x)
Sketch y = f(|x|) from y = f(x)
Distinguish the two transformations
Apply them to lines and curves

y = |f(x)| — modulus outside

For $y = |f(x)|$ , take the graph of $f$ and $\textbf{reflect any part below the x-axis up}$ above it. Everything already above stays put; the result never goes negative.

y = |f(x)|: \text{ reflect below-axis parts upwards}

Outside modulus ⟶ nothing below the x-axis.

y = f(|x|) — modulus inside

For $y = f(|x|)$ , keep the graph for $x \ge 0$ and $\textbf{reflect it in the y-axis}$ to cover $x < 0$ . The original left-hand part is discarded; the graph becomes symmetric about the $y$ -axis.

y = f(|x|): \text{ reflect the } x \ge 0 \text{ part in the y-axis}

Inside modulus ⟶ symmetric about the y-axis.

Tip — Outside = reflect bottom up (in the x-axis). Inside = reflect right side across (in the y-axis).

Telling them apart

They are genuinely different graphs. A quick check: $y = |f(x)|$ is never negative; $y = f(|x|)$ can dip below the axis but is always symmetric in the $y$ -axis.

Example 1For

f(x) = x - 2

, how do

y = |x - 2|

and

y = |x| - 2

differ?

|x-2|

: V with vertex at

(2,0)

|x|-2

f(|x|)

form... here it is a V shifted down, vertex

(0,-2)

Answer

different vertices and shapes

Formula recap

y = |f(x)|

Reflect below-axis parts up (≥ 0).

y = f(|x|)

Reflect x ≥ 0 part in the y-axis (symmetric).

|f(x)| \ge 0 \ \text{always}

Quick distinguishing check.

Common mistakes to avoid

Treating y = |f(x)| and y = f(|x|) as the same.

Outside reflects in the x-axis; inside reflects in the y-axis.

Letting y = |f(x)| go below the x-axis.

The outside modulus makes the whole graph ≥ 0.

Key takeaways

y = |f(x)|: reflect parts below the x-axis upward (never negative).
y = f(|x|): reflect the x ≥ 0 part in the y-axis (y-axis symmetric).
Check: |f(x)| ≥ 0; f(|x|) is symmetric about the y-axis.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

For $y = |f(x)|$ , you reflect parts below the x-axis:
- Up above the x-axis
- Across the y-axis
- To the right
- Down further
Show answer and explanation
Answer: Up above the x-axis
Outside modulus.
For $y = f(|x|)$ , you reflect the $x \ge 0$ part:
- In the y-axis
- In the x-axis
- In y = x
- Down
Show answer and explanation
Answer: In the y-axis
Inside modulus.
$y = |f(x)|$ is always:
- On or above the x-axis
- Symmetric in the y-axis
- Below the x-axis
- A straight line
Show answer and explanation
Answer: On or above the x-axis
Output ≥ 0.
$y = f(|x|)$ is always:
- Symmetric about the y-axis
- Never negative
- A V-shape
- One-to-one
Show answer and explanation
Answer: Symmetric about the y-axis
Built by reflecting in the y-axis.
Which can dip below the x-axis?
- $y = f(|x|)$
- $y = |f(x)|$
- Neither
- Both always
Show answer and explanation
Answer: $y = f(|x|)$
Only the outside modulus forces ≥ 0.
The modulus being OUTSIDE the function reflects in the:
- x-axis
- y-axis
- line y = x
- origin
Show answer and explanation
Answer: x-axis
Negative outputs flip up.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.