8. The modulus is the size of the number, without its sign.

Where is the vertex of the graph of y = |x − 5|?

(5, 0). The point of the V is where x − 5 = 0.

How is the graph of y = |f(x)| obtained from the graph of y = f(x)?

Reflect the parts below the x-axis in the x-axis. Parts above the axis are left alone.

Which value can y = |f(x)| never take?

−1. A modulus is never negative.

The modulus function and its graph · Cambridge IGCSE Additional Mathematics

In plain words

The modulus of a number is its size, ignoring its sign: |5| = 5 and |−5| = 5. Applied to a whole graph, it does one simple thing: anything below the x-axis is flipped up above it.

5 things to know

|x| is the modulus (absolute value) of x. It is never negative.
To draw y = |f(x)|: draw y = f(x), then reflect in the x-axis every part that lies below the x-axis. Parts above the axis stay where they are.
The graph of y = |ax + b| is a V shape, with its point on the x-axis where ax + b = 0.
For a quadratic or a cubic, the dips below the axis become humps above it.
The points where the graph meets the x-axis do not move.

Worked example

Sketch the graph of y = |2x − 4|.

Start with the line y = 2x − 4. It crosses the x-axis at (2, 0) and the y-axis at (0, −4).
The part to the left of x = 2 is below the x-axis. Reflect it in the x-axis.
The result is a V with its point at (2, 0). It crosses the y-axis at (0, 4).

Tips and tricks

Label the vertex and the intercepts on a sketch. The y-intercept changes sign if it was negative: (0, −4) becomes (0, 4).
y = |f(x)| can never be negative, so its graph never goes below the x-axis. Use that to check a sketch.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The modulus function and its graph: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is |−8|?
- −8
- 8 (the answer)
- 0
- 64
The modulus is the size of the number, without its sign.
Where is the vertex of the graph of y = |x − 5|?
- (0, 5)
- (−5, 0)
- (5, 0) (the answer)
- (0, −5)
The point of the V is where x − 5 = 0.
How is the graph of y = |f(x)| obtained from the graph of y = f(x)?
- Reflect the whole graph in the y-axis.
- Reflect the parts below the x-axis in the x-axis. (the answer)
- Move the graph up.
- Reflect the whole graph in the x-axis.
Parts above the axis are left alone.
Where does the graph of y = |2x − 6| cross the y-axis?
- (0, −6)
- (0, 6) (the answer)
- (0, 3)
- (0, −3)
At x = 0, |−6| = 6.
Which value can y = |f(x)| never take?
- 0
- 3
- −1 (the answer)
- 100
A modulus is never negative.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.

The modulus function and its graph

Cambridge IGCSE Additional Mathematics 0606 · 6 marks · papermunch.org

Name ______________________________ Date ______________

State the value of |−7.5|.[1]
Show answerHide answer
7.5
I got this one
State the coordinates of the vertex of the graph of y = |3x + 6|.[2]
Show answerHide answer
(−2, 0). The vertex is where 3x + 6 = 0.
I got this one
State the coordinates of the points where the graph of y = |x² − 4| meets the axes.[3]
Show answerHide answer
(−2, 0) and (2, 0) on the x-axis, and (0, 4) on the y-axis.
I got this one

Answers: The modulus function and its graph

1. 7.5
2. (−2, 0). The vertex is where 3x + 6 = 0.
3. (−2, 0) and (2, 0) on the x-axis, and (0, 4) on the y-axis.