Graph Transformations: Horizontal Scaling and Symmetry

Lesson 2 of 2

In this lesson:

• Compress and stretch graphs horizontally with
• Classify functions as even, odd, or neither

Learning Objectives for This Lesson

1. Identify horizontal compressions: with narrows the graph
2. Identify horizontal stretches: with widens the graph
3. Identify y-axis reflections: reflects over the y-axis
4. Define even () and odd () functions
5. Classify functions algebraically using the test

Recall: Inside Operations Change -Values

From Lesson 1 — the inside-outside rule:

• Inside : modifies the input → changes x-values → horizontal effect
• Horizontal shifts: — added inside → opposes sign of

Today: multiply inside → → also horizontal, also counterintuitive

Horizontal Scaling: Multiplying the Input

For , the input is multiplied by before applying :

• → horizontal compression (graph narrows)
• → horizontal stretch (graph widens)
• y-values are unchanged — only x-values scale by

Visual: Three Versions of

is narrower — events happen sooner. is wider — events happen later.

Why Horizontal Scaling Appears Counterintuitive

Both reach , but at different -values:

• : needs so that
• : needs so that

Double input → happens twice as fast → graph half as wide.

Horizontal Reflection: Flipping Graphs Over the Y-Axis

replaces with — the input changes sign:

• Every point on becomes on
• Reflection over the y-axis

Example: → , domain becomes

Worked Example: Identifying a Horizontal Compression

From , identify the transformation to :

• inside → horizontal compression by factor

Graph is narrower than . Same vertex, faster growth.

Worked Example: Reflection Over the -Axis

From (domain ) to :

• → reflection over the y-axis
• Domain changes from to

The curve appears on the left side of the y-axis.

Quick Check: Compression or Stretch?

. For :

• reaches at , while reaches at
• gets there 3 times faster — so is...

Find . Is horizontally compressed or stretched?

Guided Practice: Identify the Horizontal Stretch

From to :

• (what's multiplying inside?)
• Is this a stretch or compression?
• By what factor does the graph widen?

Try each step, then advance for the answer.

Practice: Identifying Horizontal Scaling Transformations

Identify each transformation from :

1. — compress by (narrows)
2. — stretch by 2 (widens)
3. — no change (even function)
4. from — reflection over y-axis

Symmetry: What Happens When We Try ?

Computing reveals symmetry:

• → — unchanged!
• → — negated!

When the result is always → even function.
When the result is always → odd function.

Even Functions and Y-Axis Symmetry Defined

A function is even if for all in the domain:

• Graph is symmetric about the y-axis
• Each point has a mirror image

Odd Functions and Origin Rotational Symmetry

A function is odd if for all in the domain:

• Graph is symmetric about the origin
• Each point has a rotational image

Worked Example: Test

Is even, odd, or neither?

Step: Compute :

Since for all : even function.

Worked Example: Two Contrasting Cases

Case 1: . Compute → odd

Case 2: . Compute

and → neither

Quick Check: Classifying the Absolute Value Function

Compute . Recall for all .

So .

Is even, odd, or neither?

Even, Odd, and Neither: Examples Gallery

Most functions are neither even nor odd.

Practice: Classify as Even, Odd, or Neither

Key Takeaways: Scaling and Function Symmetry

✓ : horizontal — compresses, stretches
✓ : reflects over the y-axis
✓ Even: — y-axis symmetry
✓ Odd: — origin symmetry

compresses — not stretches
Even ≠ even exponents — always use the test

Next Steps: Building on Transformations

Upcoming: Inverse Functions (HSF.BF.B.4)

The graph of is the reflection of over .

Full transformation toolkit: