Integrated Math 2–3 Honors · Lesson Review · August 20, 2026

Piecewise Definitions and Transformations of Absolute-Value Functions

Turn absolute value into two linear pieces, graph the parent function, transform it with vertex form, and write an equation from a V-shaped graph.

37-minute edited lesson9 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Piecewise Definitions 0:4302Example: |x − 3|3:4703More Piecewise Examples 6:2804The Parent Absolute-Value Function 10:5105Horizontal Translations and the Vertex 17:5906Vertex Form and Transformations 23:4407Vertical Stretch and Compression 31:0508Reflections Across the x-axis 35:0509Write an Equation from a Graph

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02 · Understand

Learning targets

Write an absolute-value expression as a piecewise-defined function.
Graph $y=|x|$ from its two linear pieces.
Graph $y=a|x-h|+k$ using its vertex, slopes, and transformations.
Write an absolute-value equation from a V-shaped graph.

03 · Vocabulary

Key definitions

Use these words to describe what changes—and what stays the same—when a graph is transformed.

01Absolute-value function: A function whose rule contains an absolute-value expression. Its graph is usually V-shaped.
02Piecewise-defined function: A function described by different formulas on different parts of its domain.
03Parent function: The simplest function in a family. For absolute value, the parent is $y=|x|$ .
04Vertex: The turning point where the two rays of an absolute-value graph meet.
05Translation: A slide that moves a graph left, right, up, or down without changing its shape.
06Vertical stretch or compression: A change in the graph’s steepness caused by multiplying every y-value by a factor.
07Reflection across the x-axis: A transformation that changes every y-value to its opposite, turning the V upside down.
08Vertex form: The form $y=a|x-h|+k$ , where the vertex is $(h,k)$ and $a$ controls slopes and orientation.

04 · Apply

Worked examples

Start with the inside expression, then read the vertex and slopes.

Piecewise rule

Rewrite $|x-3|$

|x-3|=\begin{cases}x-3,&x\ge3,\\-x+3,&x<3.\end{cases}

The switch happens where the whole inside is zero: $x-3=0$ , so $x=3$ .

Negative coefficient

Find the boundary for $|6-2x|$

$6-2x\ge0$
$-2x\ge-6$
$x\le3$ reverse the inequality when dividing by −2

Parent graph

Graph $y=|x|$

y=\begin{cases}x,&x\ge0,\\-x,&x<0.\end{cases}

The two rays meet at $(0,0)$ . The right-hand slope is 1; the left-hand slope is −1.

Translation

Graph $y=|x-2|-4$

Move right 2 and down 4.
Vertex: $(2,-4)$ .
From the vertex, rise 1 and run 1 in both directions.

Stretch

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

Vertex: $(-1,-2)$ .
The graph opens upward because $a=3>0$ .
Use slopes 3 and −3: rise 3 for every run of 1.

Compression

Graph $y=\frac12|x-3|+1$

The vertex is $(3,1)$ . Because $|a|=\frac12$ , the V is wider: rise 1 for every run of 2.

Reflection

Graph $y=-4|x+2|+5$

The vertex is $(-2,5)$ . The negative coefficient reflects the graph across the x-axis, so it opens downward with slopes −4 and 4.

From a graph

Write the equation

A V has vertex $(-3,1)$ , opens upward, and rises 2 for a run of 1.

y=2|x+3|+1

05 · Avoid

Common mistakes

Testing only x against zero.

The sign test belongs to the entire expression inside the absolute value.

Forgetting to distribute the negative.

For example, $-(x-3)=-x+3$ .

Not reversing an inequality.

Reverse the symbol whenever you multiply or divide by a negative number.

Moving in the visible inside direction.

$x+3$ shifts left 3; $x-2$ shifts right 2.

Using a as the vertex.

The vertex comes from h and k. The value a controls steepness and orientation.

Missing the reflection.

If $a<0$ , the graph opens downward.

06 · Check yourself

Quick check

Decide first, then open the answer.

1Where does the rule for

|2x+4|

switch?

At $x=-2$ . Solve $2x+4=0$ .

2What is the vertex of

y=|x+5|-2

$(-5,-2)$ . The inside sign is opposite; the outside number keeps its sign.

3Describe

y=-2|x-3|

Vertex $(3,0)$ , opens downward, and has slopes −2 and 2.

4Which graph is wider:

y=3|x|

y=\frac12|x|

$y=\frac12|x|$ . A smaller absolute value of $a$ makes the V less steep.

5A downward V has vertex

(4,2)

and slope magnitude

\frac23

. Write its equation.

$y=-\frac23|x-4|+2$ .

07 · Revisit

Transcript and assigned practice

Search the corrected transcript for a definition or example. Then complete the eKadence worksheet Graphing Absolute Value Equations.

Download corrected transcript