The modulus functionAQA A-Level Maths: Revision notes

The modulus of a number is its distance from zero, so it is never negative: $|a|=a$ if $a\ge0$ and $|a|=-a$ if $a<0$. For example $|5|=5$ and $|-5|=5>0$. For a linear function, $y=|ax+b|$ means $y=ax+b$ when $ax+b\ge0$ and $y=-(ax+b)$ when $ax+b<0$. The output of a modulus function is therefore always $\ge0$, so the range of $y=|ax+b|$ is $y\ge0$.

Method:

1. Sketch the straight line $y=ax+b$, marking where it crosses both axes.
2. Keep every part that is on or above the $x$-axis.
3. Reflect every part below the $x$-axis in the $x$-axis. The result is a V-shape whose lowest point, the vertex, lies on the $x$-axis at $\left(-\frac ba,0\right)$. The two branches have gradients $a$ and $-a$.

• Vertex: solve $ax+b=0$, giving $\left(-\frac ba,0\right)$.
• $y$-intercept: put $x=0$ to get $y=|b|$, always positive.
• Line of symmetry: the vertical line $x=-\frac ba$ through the vertex.
• Piecewise form: one branch is $y=ax+b$, the other is $y=-(ax+b)$, joined at the vertex. Example: $y=|2x-6|$ has vertex $(3,0)$, $y$-intercept $6$, symmetry line $x=3$, and equals $2x-6$ for $x\ge3$, $6-2x$ for $x<3$.

The curve $y=|2x-5|$ has vertex $\left(\frac52,0\right)$ and $y$-intercept $(0,5)$. To find where it meets $y=x+1$, solve each branch separately:

• Right branch ($x\ge\frac52$): $2x-5=x+1$, so $x=6$. Valid, since $6\ge\frac52$.
• Left branch ($x<\frac52$): $5-2x=x+1$, so $x=\frac43$. Valid, since $\frac43<\frac52$. Always check that each solution lies in the interval for its branch.

The horizontal line $y=k$ meets $y=|ax+b|$ twice if $k>0$, once if $k=0$ (at the vertex) and never if $k<0$. For a sloping line $y=mx+c$, compare gradients. If $|m|<|a|$ the line is shallower than both branches, so it meets the V twice when it passes above the vertex, once when it passes through the vertex and never when it passes below. Substituting the vertex into the line gives the boundary value. Example: $y=x+k$ against $y=|2x-5|$ passes through the vertex when $0=\frac52+k$, so $k=-\frac52$.