Transformations of Power Functions With Positive Integer Index (VCE SSCE Mathematical Methods): Revision Notes

Transformations of Power Functions With Positive Integer Index

Introduction to power functions

A power function is a function with the form $f(x) = x^n$, where $n$ is a positive integer. These functions form the building blocks for many polynomial expressions.

Not all polynomials can be written in transformation form, but many can be expressed as:

$$y = a(x - h)^n + k$$

where $a$, $h$, and $k$ are real constants.

Power functions with odd positive integer index

Basic characteristics of odd power functions

When $n$ is an odd positive integer (such as $3, 5, 7, ...$), the function $f(x) = x^n$ has distinctive properties that set it apart from other function types.

Key features of odd power functions:

• The graph passes through the origin $(0, 0)$
• The graph has an S-shaped curve
• The function is always increasing
• The graph extends from negative infinity to positive infinity

Graphs of odd power functions

The diagrams below show the fundamental shapes of $y = x^3$ and $y = x^5$.

Notice that both graphs:

• Pass through the origin
• Have the same general S-shape
• Are steeper as we move away from the origin
• Show marked points demonstrating the rapid growth of higher powers

Comparing odd power functions

When we compare two odd power functions where $n > m$ (both odd), several important relationships emerge that help us understand their relative behavior.

The appearance of these graphs depends on the scale used on the axes. A typical representation of an odd power function shows the characteristic S-curve:

Derivative properties of odd power functions

For $f(x) = x^n$ where $n$ is an odd integer with $n \geq 3$, the derivative is:

$$f'(x) = nx^{n-1}$$

This derivative tells us about the gradient (slope) of the function. Since $n$ is odd, $n - 1$ is even, which has important implications:

• The gradient is zero when $x = 0$
• Therefore $f'(x) = nx^{n-1} > 0$ for all non-zero $x$

Transformations of odd power functions

Transformations of $f(x) = x^n$ (where $n$ is odd) result in graphs with rules of the form:

$$y = a(x - h)^n + k$$

where:

• $a$ affects vertical stretching/compression and reflection
• $h$ represents horizontal translation
• $k$ represents vertical translation

The point of zero gradient moves from $(0, 0)$ to $(h, k)$.

Worked example: Sketching transformed cubic functions

Worked example: Finding parameters

Power functions with even positive integer index

Basic characteristics of even power functions

When $n$ is an even positive integer (such as $2, 4, 6, ...$), the function $f(x) = x^n$ has these properties that create a fundamentally different shape from odd power functions:

• The graph has a U-shape (parabola-like)
• There is a minimum point at the origin $(0, 0)$
• The graph is symmetric about the $y$-axis
• The function decreases for $x < 0$ and increases for $x > 0$

Comparing even power functions

When we compare two even power functions where $n > m$ (both even), we observe similar patterns to odd functions but with important differences.

Here's a comparison showing the different shapes of odd versus even power functions:

Derivative properties of even power functions

For $f(x) = x^n$ where $n$ is an even integer with $n \geq 2$, the derivative is:

$$f'(x) = nx^{n-1}$$

Since $n$ is even, $n - 1$ is odd. This means:

• The gradient is zero when $x = 0$
• $f'(x) = nx^{n-1} > 0$ for $x > 0$ (function is increasing)
• $f'(x) = nx^{n-1} < 0$ for $x < 0$ (function is decreasing)

Transformations of even power functions

Similar to odd power functions, transformations of $f(x) = x^n$ (where $n$ is even) produce graphs with the form:

$$y = a(x - h)^n + k$$

The turning point moves from $(0, 0)$ to $(h, k)$.

Worked example: Finding parameters for even functions

Remember!