Taking the Modulus of a Whole Expression

Start from the curve

Take the parabola $y=x^2-4$y = x² − 4. Its lowest point, the vertex, is (0, −4), because $x^2$x² is smallest at x = 0. It crosses the x-axis where $x^2-4=0$x² − 4 = 0, so where $x^2=4$x² = 4, which is at x = 2 and x = −2.

Between those two crossings the curve is below the x-axis. At x = 1, for example, y = 1 − 4 = −3. Outside them it is above the axis: at x = 3, y = 9 − 4 = 5.

$y=x^2-4$y = x² − 4 crosses the x-axis at −2 and 2, and between them it dips below the axis to its vertex at (0, −4).

What the bars do to each value

Now put bars around the whole expression: $y=|x^2-4|$y = |x² − 4|. To find a value, work out $x^2-4$x² − 4 first, and then take its size.

Where $x^2-4$x² − 4 is already positive, the bars change nothing: at x = 3 it is 5, and |5| = 5. Where it is negative, the bars turn it into the positive number of the same size: at x = 0 it is −4, and |−4| = 4. Where it is zero, at x = −2 and x = 2, the value stays 0.

The positive values and the zeros are unchanged. Each negative value becomes the positive number of the same size.

Reflect the part below the axis

On the graph, a value that changes from −4 to 4 is a point that moves from (0, −4) to (0, 4): the same distance from the x-axis, on the other side of it. So the part of the parabola below the x-axis is reflected in the x-axis, and the parts above the axis stay where they are.

The dip between −2 and 2 turns into a hump, with its top at (0, 4). The whole graph of $y=|x^2-4|$y = |x² − 4| is on or above the x-axis, and its smallest value is 0, reached at x = −2 and x = 2.

The gold graph is $y=|x^2-4|$y = |x² − 4|. The white curve is the dip of $y=x^2-4$y = x² − 4 below the axis, which the modulus reflects up into a hump with its top at (0, 4).

A corner at each crossing

At x = −2 and x = 2 the graph comes down to the x-axis and goes straight back up. It touches the axis there without crossing it, because no part of the graph is below the axis.

The turn is sudden. Just to the left of x = 2, the reflected curve slopes down to (2, 0) as steeply as the original curve climbs there, and just to the right the original curve climbs away. The two pieces meet in a sharp point, a corner, and not in a smooth bend like the vertex of a parabola. The same happens at x = −2.

$y=|x^2-4|$y = |x² − 4| touches the x-axis at (−2, 0) and (2, 0), with a corner at each.

The same rule for any graph

To draw the modulus of any expression, draw the graph of the expression first. Keep every part that is on or above the x-axis, and reflect every part that is below it in the x-axis.

Applied to the straight line y = 2x − 4, the rule gives a V. The line is below the axis to the left of x = 2, so that part is reflected up, and y = |2x − 4| is a V with its corner at (2, 0).

Do not confuse this with putting the bars around x alone, as in $y=x^2-4|x|$y = x² − 4|x|. That is a different operation, and it gives a different picture.

The white line is the part of y = 2x − 4 below the x-axis. Reflected up, it becomes the left arm of the gold V, y = |2x − 4|, which turns at (2, 0).

The usual mistakes

Keeping the value from before the bars. At x = 0, $x^2-4=-4$x² − 4 = −4, but $y=|x^2-4|$y = |x² − 4| is 4. A modulus graph never goes below the x-axis.

Reflecting the whole curve. Only the part below the axis moves. At x = 3, $x^2-4=5$x² − 4 = 5 is already above the axis, so the value stays 5; reflecting everything would give −5.

Giving the smallest value as −4 or 4. −4 was the lowest point before the bars. 4 is the top of the hump, and the graph falls away from it on both sides, down to 0 at x = −2 and x = 2. The smallest value is 0.