Lesson 5.15

Steepness

Some hills are easy to walk up. Some are cliffs. In algebra, we control this intensity with a simple multiplier.

Introduction

We've shifted lines up and down. We've flipped them over. Now we are going to stretch them out or squash them flat. This is called Vertical Stretch/Compression, but we often just talk about "Steepness."

Past Knowledge

Lesson 5.1 (Slope). You recall that big numbers mean "steep" slope.

Today's Goal

Compare function to .

Future Success

This is crucial for Parabolas. is a wide bowl. is a thin spike.

Key Concepts

Vertical Stretch & Compression

Stretch (Steeper)

If , the graph gets steeper (pulling away from the x-axis).

Example: is very steep.

Compression (Flatter)

If , the graph gets flatter (pushed toward the x-axis).

Example: is gentle.

Note: Ignore the negative sign when deciding steepness. is just as steep as . Look at the Absolute Value.

Worked Examples

Example 1: Identifying Change

Basic

Compare to .

Step 1: Identify "a"

The multiplier is 4.

Step 2: Classify

Is 4 bigger than 1? Yes.

Vertical Stretch by a factor of 4.

It is 4 times steeper.

Example 2: Fractions

Intermediate

Compare to .

Step 1: Identify "a"

The multiplier is .

Step 2: Classify

Is it less than 1? Yes.

Vertical Compression by a factor of 1/3.

It is flatter.

Example 3: Combined Transformation

Advanced

Describe .

Analysis

There are two things happening:

Negative sign: Reflection over x-axis.
Number 5: Vertical Stretch by factor of 5.

Reflect and Stretch.

Common Pitfalls

Fraction vs Negative

Students confuse "negative number" with "compression".
is a STRETCH because .
is a COMPRESSION.

Steaming Rolling vs Pulling

"Compression" sounds like squashing it flat (towards x-axis). "Stretch" sounds like pulling it tall (away from x-axis). Visualizing taffy or play-dough helps.

Real-Life Applications

Speed Multipliers: If represents driving at 30mph, then represents driving TWICE as fast. Wait, that's horizontal... actually represents "Going twice as far in the same time". That's a vertical stretch!