1.4 Modulus Function Graphs

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📖 Main Content

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What Does the Modulus Mean? (Quick Recap)

The modulus of a number simply means its size, ignoring whether it is positive or negative. It is always written using two vertical bars, like this: | |

Think of it as the distance from zero on a number line — distance is never negative.

• |5| = 5 (already positive, so stays the same)
• |−5| = 5 (negative, so we remove the minus sign)
• |0| = 0

So the modulus of any number is always zero or positive — never negative.

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The Core Rule: From y = f(x) to y = |f(x)|

When you write y = |f(x)|, you are applying the modulus to the entire output of the function f(x). This changes the graph in one specific way:

This works because:

• When f(x) is positive or zero, |f(x)| = f(x) → no change
• When f(x) is negative, |f(x)| = −f(x) → the negative is removed, so the y-value becomes positive

Think of it like this: the x-axis acts like a mirror. Any part of the curve that dips below the mirror gets flipped back up. The shape of those flipped parts stays the same — only their position changes (from below the x-axis to above it).

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Step-by-Step Process for Sketching y = |f(x)|

Follow these steps every time:

1. Sketch y = f(x) first — draw the original graph, including all key points (intercepts, turning points, etc.)
2. Identify which parts are below the x-axis — these are the sections where f(x) < 0
3. Reflect those parts in the x-axis — flip them upward
4. Keep everything above the x-axis exactly as it is
5. Mark all key coordinates on your final sketch