2.1, 3.1 - Modulus graphs and equations

Distance from zero

Which inputs make $2x-1$ exactly as far from zero as $x+2$? Before solving that puzzle, we need a way to describe distance without a sign. Both $5$ and $-5$ are five units from zero, so $|5|=|-5|=5$.

For $y=|2x-4|$, first find where the inside changes sign: $2x-4=0$ gives $x=2$. To the right, keep the line $2x-4$. To the left, negate it to make its output positive. The two rays meet at the vertex, $(2,0)$.

Change only the sign of the coefficient of $x$. Find the new zero rather than guessing where the V moves.

Both graphs open upwards: changing the sign inside a modulus does not produce negative distances.

[DIAGRAM: asset_name: Splitting a line at its zero; asset_slug: modulus-linear-pair; caption: The zero of the expression gives the vertex of each V-shaped graph; description: Two equally scaled coordinate plots show y equals modulus of 2x minus 4, with vertex 2 comma 0, and y equals modulus of minus 2x minus 4, with vertex minus 2 comma 0. Both have y-intercept 0 comma 4. Each left ray has gradient minus 2 and each right ray gradient 2.]

For $a\ne0$, $y=|ax+b|$ has vertex $(-b/a,0)$, $y$-intercept $(0,|b|)$, and ray gradients $-|a|$ and $|a|$. If $a=0$, the graph is the horizontal line $y=|b|$, rather than a V.

Locate a new vertex and write the two rules.

Equating two distances

A distance of $5$ can come from $5$ or $-5$. Thus $|u|=5$ means $u=5$ or $u=-5$. More generally, two real numbers have equal moduli exactly when they are equal or opposite.

The signed cases usually avoid expanding squares. If an equation instead has $|u|=v$, the right side must be non-negative; squaring alone can otherwise introduce false answers.