Graphs of modulus (absolute value) functions

Taking the modulus reflects every part of a graph that lies below the x-axis up above it. That gives the sharp V shapes and the corners these three graphs share.

Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).

Graph of y = |x|

The modulus (absolute value) graph: a V shape with its vertex at the origin. For x < 0 it is y = −x, for x ≥ 0 it is y = x.

Open y = |x| in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ 0
x-intercept	(0, 0)
y-intercept	(0, 0)
Turning point	minimum (0, 0), a corner (the gradient is not defined there)
Asymptotes	none
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercept (0, 0), the y-intercept (0, 0), the turning point (0, 0), a corner (the gradient is not defined there).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ 0. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

Graphs made from this one: y = |x − 2|. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = |x − 2|

The V-shaped graph of y = |x| moved 2 to the right.

Open y = |x − 2| in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ 0
x-intercept	(2, 0)
y-intercept	(0, 2)
Turning point	minimum (2, 0), a corner
Asymptotes	none
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercept (2, 0), the y-intercept (0, 2), the turning point (2, 0), a corner.
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ 0. Your sketch should not go anywhere else.

y = |x − 2| is y = |x| translated 2 units to the right: see the graph of y = |x|. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = |x² − 4|

The part of y = x² − 4 below the x-axis is reflected above it, giving a W shape with corners at x = ±2.

Open y = |x² − 4| in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ 0
x-intercepts	(−2, 0) and (2, 0)
y-intercept	(0, 4)
Turning points	minimum (−2, 0), a corner; maximum (0, 4); minimum (2, 0), a corner
Asymptotes	none
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercepts (−2, 0) and (2, 0), the y-intercept (0, 4), the turning points (−2, 0), a corner and (0, 4) and (2, 0), a corner.
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ 0. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

y = |x² − 4| is y = x² − 4 with the part below the x-axis reflected in the x-axis: see the graph of y = x² − 4. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

The Grapher is our own free tool (graph.mathrevision.co.uk), built for IB, A Level and IGCSE students. It is not an exam calculator: practise on your own GDC for calculator papers.