Lesson 4.2

Power Functions and Their Behavior

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

Introduction

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

Past Knowledge

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

Today's Goal

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).

Future Success

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

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-Life Applications

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.