Number of Solutions from a Modulus Graph

To count the solutions of a modulus equation, sketch the curve and the line, then count where they cross. Each intersection is one solution. Here $y = |x^2 - 4|$ is a “W” resting on the $x$-axis.

By sketching suitable graphs, find the number of solutions of $|x^2 - 4| = 3$. [4]

The working

Step 1, sketch $y = |x^2 - 4|$. The parabola $y = x^2 - 4$ dips below the axis between $x = -2$ and $x = 2$; the modulus reflects that dip upward, giving a “W” with the middle peak at $(0, 4)$ and two minima on the $x$-axis at $(\pm 2, 0)$. (M1, A1)

Step 2, draw the line $y = 3$ across the sketch. (M1)

Step 3, count intersections. The line $y = 3$ cuts:

• the two outer arms once each (where $x^2 - 4 = 3$),
• the central hump twice (where $4 - x^2 = 3$).

That is four intersection points, so four solutions. (A1)

Check algebraically: $x^2 - 4 = 3 \Rightarrow x = \pm\sqrt{7}$; $\ 4 - x^2 = 3 \Rightarrow x = \pm 1$. Four values. ✓

Where the marks are won and lost

• The graph of $|x^2 - 4|$ is a “W”: the part between $x = \pm 2$ flips above the axis, creating the central peak at $(0,4)$.
• The line $y = 3$ sits below the peak ($3 < 4$), so it cuts the hump twice; a line above $4$ would miss the hump entirely.
• Count all crossings, including both arms and both sides of the hump.

Common mistakes

• Forgetting the reflected central hump and counting only two solutions.
• Drawing $y = 3$ above the peak and getting the count wrong.
• Solving only $x^2 - 4 = 3$ and missing $4 - x^2 = 3$.

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