Lesson 4.2

Power Functions and Their Behavior

From simple lines to steep curves. All polynomials are built from these LEGO bricks.

Introduction

Prerequisite Connection: We've seen (linear) and (quadratic). What about , , or ?

Today's Increment: We define a Power Function as where is a positive integer. We'll discover that they essentially fall into two families: Evens (like parabolas) and Odds (like s-curves).

Why This Matters for Calculus: Taylor Series (Calculus II) allows us to approximate ANY function (even ugly ones like ) as a sum of simple power functions.

Explanation of Key Concepts

The Two Families

Even Degree (x², x⁴, x⁶...)

Shape: U-shaped (Parabolic).
Symmetry: Across Y-axis (Even Function).
End Behavior: Ends point in SAME direction (both UP or both DOWN).

Odd Degree (x³, x⁵, x⁷...)

Shape: S-shaped (Chair).
Symmetry: About Origin (Odd Function).
End Behavior: Ends point in OPCposite directions (one UP, one DOWN).

Worked Examples

Level: Conceptual

Example 1: The "Flatness" of Higher Powers

Compare and .

Near Zero (-1 to 1):

is flatter/smaller. (e.g.,

Outside (-1 to 1):

grows WAY faster. Steep walls.

Level: Intermediate

Example 2: Negative Odd Power

Analyze .

The negative sign reflects the S-curve over the x-axis.

Original: Low on left, High on right.
Reflected: High on left, Low on right.

Level: Advanced

Example 3: Solving Power Inequalities

For what values of x is ?

Step 1: Set to Zero

Step 2: Factor

Step 3: Test Intervals

Rational roots at -1, 0, 1.

(-∞, -1): Neg * Pos = Neg
(-1, 0): Neg * Neg = Pos
(0, 1): Pos * Neg = Neg
(1, ∞): Pos * Pos = Pos

Answer:

Common Pitfalls

Confusing Big X with Small X:
Just because is "stronger" doesn't mean it's always bigger. Between -1 and 1, higher powers are SMALLER. This is crucial for convergence tests in Calculus.
Assumptions about Negatives:
(Positive).
(Negative).
The Order of Operations matters immensely with even powers!

Real-World Application

Biology: Allometry

The relationship between an animal's body mass and metabolic rate follows a power law.

Kleiber's Law states . This means if you are 10,000 times heavier (like a mouse vs an elephant), you actulaly need less energy per pound than the mouse does. Power functions govern the scaling of life.