Absolute Value Functions: Graph the V on a Live Board

Absolute value functions take each input to its distance from zero. The graph of

$$y=|x|$$

is a V: the vertex sits at the origin. The right arm rises with slope $1$. The left arm falls with slope $-1$ as $x$ increases through negative values — both arms meet at $(0,0)$.

Parent shape — $y=|x|$

Pick a few inputs. At $x=0$ the height is $0$. At $x=2$ the height is $2$. At $x=-2$ the height is also $2$, because absolute value forgets the sign. The two arms meet at the vertex $(0,0)$. That meeting point is what later shifts when $h$ and $k$ move.

Reading the graph as pairs $(x,f(x))$ is the same habit as on any real function: each input gets one height. Absolute value just makes both sides of zero land on the same nonnegative height, so the picture folds into a V instead of a single slanted line.

Transforms — $y=a|x-h|+k$

The same three numbers used for other parents also move this V:

The transformed graph is

$$y=a|x-h|+k.$$

Gray dashed is the parent $y=|x|$. Green is $y=a|x-h|+k$. Drag the sliders for $a$, $h$, and $k$ and watch the green V move against the gray parent.

Notation note. Classroom $a$, $h$, and $k$ are the same moves the handbook writes with $a$ and $b$ on transformation of functions. A shift there is $y=f(x-a)+b$, so handbook $a$ is this post’s $h$ and handbook $b$ is this post’s $k$. A positive stretch there is $y=bf(x/a)$. This post keeps $a$, $h$, and $k$ so the three sliders match what many worksheets print for absolute value. The handbook page keeps its own letters.

Worked example — from $y=|x|$ to $y=2|x-1|+3$

Parent: $f(x)=|x|$ (gray dashed).

Target:

$$y=2|x-1|+3.$$

Read the three numbers: $a=2$, $h=1$, $k=3$.

1. Stretch by $2$: $y=2|x|$ is steeper than $y=|x|$.
2. Shift right by $1$: replace $x$ with $x-1$ → $y=2|x-1|$.
3. Shift up by $3$: add $3$ → $y=2|x-1|+3$.

The vertex of the parent is $(0,0)$. After these steps the vertex sits at $(1,3)$. Check one point: at $x=1$, $y=2|0|+3=3$. At $x=2$, $y=2|1|+3=5$. Set the sliders to $a=2$, $h=1$, $k=3$ and match that green V to the algebra — the vertex should sit clear of the slider strip near the top of the board.

Reflect. If $a=-2$ instead, the same shifts give $y=-2|x-1|+3$: the V opens downward (flip over the x-axis) with vertex still at $(1,3)$. Negative $a$ is still the same parameter — stretch size $|a|$, then flip.

Drag the sliders

Gray dashed = parent $y=|x|$. Green = $y=a|x-h|+k$.

• Drag $a$: arm steepness and opening direction change; the vertex’s $x$-coordinate stays put until you move $h$.
• Drag $h$: the whole green V slides left or right.
• Drag $k$: the whole green V slides up or down.

That is the check that the figure matches the formula: one parent V, three numbers, and every green point is the absolute-value rule after stretch/flip and shift.

Same moves, different parent

The handbook page on transformation of functions uses the same shift-and-stretch idea (with its own letters). The post on function transformations walks $a$, $h$, and $k$ on a parabola parent $y=x^2$. Here the parent is the absolute-value V. For reading a graph as pairs $(x,f(x))$, see real functions and their graphs. If you are still building the habit of plotting pairs before you transform them, how to graph a function is the lighter step.

Same V shape. Only $a$, $h$, and $k$ change where it sits and how tall the arms are — GoodMaith keeps that motion on a live board.