Piecewise Transitions

Absolute Value Symmetry & Piecewise Logic

Name

Date

01. The Piecewise Foundation

Absolute value functions are just two linear functions meeting at a turning point. Define the parent function piecewise:

\[ f(x) = |x| = \begin{cases} \text{\underline{\hspace{3cm}}} & \text{if } x < 0 \\[15pt] \text{\underline{\hspace{3cm}}} & \text{if } x \ge 0 \end{cases} \]

Sketch the parent graph here
(Label the vertex)

02. Video Analysis

Example 1

\(y = |x - 2| + 3\)

Vertex

Axis of Symmetry

Example 2

\(y = 4 - |x - 3|\)

Vertex

Axis of Symmetry

Example 3 (Complex)

\(y = |3x - 7| + 4\)

Vertex

Axis of Symmetry

03. The Piecewise Translation

Goal: Rewrite Example 3 as a formal piecewise function.

\( y = |3x - 7| + 4 \)

Step 1: Find the Turning Point (AoS)

Solve for where the internal expression equals zero.

Step 2: Define the Domains

The function splits at the x-coordinate of the vertex.

Left Branch \( x < \) ________

Right Branch \( x \ge \) ________

Step 3: Simplify the Branches

Multiply the inside by \(-1\) for the negative branch, and \(+1\) for the positive branch. Then combine like terms.

BRANCH A: \( x < 7/3 \)

BRANCH B: \( x \ge 7/3 \)

Final Piecewise Definition

\[ f(x) = \begin{cases} \text{\underline{\hspace{5cm}}} & \text{if } x < \frac{7}{3} \\[20pt] \text{\underline{\hspace{5cm}}} & \text{if } x \ge \frac{7}{3} \end{cases} \]

04. Calculus Preview: Slopes

Calculate the slope (\(m\)) for each simplified linear branch from Step 3:

Slope of Left Branch:

Slope of Right Branch:

Critical Thinking

In calculus, the derivative describes the slope at a single point. If you approach the vertex from the left, you get one slope. If you approach from the right, you get another. Is it possible to define a single, unique slope at exactly \(x = 7/3\)? Why or why not?

Absolute Symmetry Piecewise Transitions

Absolute Symmetry Lesson Plan

Piecewise Transitions Worksheet

Piecewise Transitions Answer Key

Absolute Symmetry Slides

Similar Lessons

Vector Voyages

Erin Esswein

Love Stakes Game Show

Kelly Brown

Data Detectives

egambale

Data Detectives

Erin Esswein

Data Detective Lab

Erin Esswein

Courbe en Cloche知识

Hugo Pierrard

Art de la Visualisation知识

Hugo Pierrard

Tendance et Dispersion知识

Hugo Pierrard

Nature des Données知识

Hugo Pierrard

Basic Ops Blueprint

Carmela Leone

Math Mastery

Katie Heck

Quadratic Factor Quest

Gregory Jordan

Blueprint Builders Polynomial Addition

Cheydon Krysl

Wave Navigator

Abha Poddar

Conversion Command

Carmela Leone

Function Foundations

Jelani Arnold

Parabola Talk

Quan Nguyen

Linear Language

Grant Johnston