Pre-Calculus: Transformations

Scaling Sleuths

Cracking the code of horizontal and vertical transformations.

Today's Investigation

1 The Goal

Distinguish between horizontal and vertical scaling through coordinate analysis.

2 The Evidence

Analyze invariant points—the points that refuse to move when a graph is stretched or squished.

Warm-up: The Parabola Case

Open Desmos or your graphing calculator and graph:

\(y = x^2\)

The Original (Parent)

vs.

\(y = 2x^2\)

The Transformation

What changed visually? Did it get skinnier or wider?

Look at the point (0, 0). Did it move?

Case Study 1: Absolute Value

Watch for Randy's logic: Why does he rule out vertical scaling?

The Vertex Sticky-Note

"The vertex at (0, 1) did not move up or down."

In a vertical scaling, every y-value is multiplied by a constant \(k\).

\(1 \times k \neq 1\)

(unless k = 1, which isn't a stretch!)

The Logic:

Vertex is at \((0, y)\) where \(y \neq 0\).
If \(y\) doesn't change, it's NOT vertical.
Since \(x = 0\), horizontal scaling does nothing!

Case Study 2: Sine Waves

Check the intercepts: Why does he rule out horizontal scaling?

The Intercept Lockdown

"The roots line up at 0, 3, etc. on both curves."

In a horizontal scaling, every x-value is multiplied by a constant \(1/b\).

\(3 \times (1/b) \neq 3\)

(unless b = 1)

The Verdict:

Intercepts are at \((x, 0)\) where \(x \neq 0\).
If \(x\) doesn't change, it's NOT horizontal.
Height changed, so it MUST be vertical!

Summary: The Invariant Points

Point Type	Vertical Scale	Horizontal Scale
\(y\)-intercepts \((0, y)\)	Moves! \(y\) changes	STAYS FIXED \(x=0\) doesn't change
\(x\)-intercepts \((x, 0)\)	STAYS FIXED \(y=0\) doesn't change	Moves! \(x\) changes

FINISHING UP

Stretch or Squish?

Pick up your Exit Ticket and analyze the three crime scenes. Can you identify the scaling type for each?

Analyze Points

Justify Logic

Submit & Go

Scaling Sleuths

Scaling Sleuths Slides

Stretch or Squish Exit Ticket

Scaling Sleuths Teacher Guide

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