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 Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.5 Sketching Modulus FunctionsWatch the full walkthrough 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 $reflect any part below the x-axis up$ above it. Everything already above stays put; the result never goes negative.

y = ∣ f (x) ∣ : 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 $x \geq 0$ and $reflect it in the y-axis$ to cover $x < 0$ . The original left-hand part is discarded; the graph becomes symmetric about the $y$ -axis.

y = f (∣ x ∣) : reflect the x \geq 0 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.

∣ x - 2∣

: V with vertex at

(2, 0)

∣ x ∣ - 2

f (∣ x ∣)

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

(0, - 2)

Answerdifferent 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) ∣ \geq 0 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

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