Deciphering the Nuances of Even-Powered Functions: A Detailed Analysis of x^2, x^4, (x-1)^2, and (x-1)^4

Behavior Close to x=0 for x²

The function f(x) = x² behaves quadratically around x = 0. As you approach 0 from either the left or the right, the function also approaches 0.

• As x approaches 0 from the left, f(x) approaches 0 from above.
• As x approaches 0 from the right, f(x) approaches 0 from above.

This behavior indicates that the root at x = 0 has a multiplicity of 2.

Because the root has a multiplicity of 2, the graph of the function “bounces” off the x-axis at x = 0.

Table of Calculations Close to x=0 for x²

Behavior Close to x=0 for x⁴

The function f(x) = x⁴ behaves in a quartic manner around x = 0. As you approach 0 from either the left or the right, the function also approaches 0.

• As x approaches 0 from the left, f(x) approaches 0 from above.
• As x approaches 0 from the right, f(x) approaches 0 from above.

This behavior indicates that the root at x = 0 has a multiplicity of 4.

Because the root has a multiplicity of 4, the graph of the function “bounces” off the x-axis at x = 0, similar to x² but with a more pronounced flattening.

Table of Calculations Close to x=0 for x⁴

Behavior Close to x=1 for (x-1)²

The function f(x) = (x-1)² behaves quadratically around x = 1, similar to how f(x) = x² behaves around x = 0.

• As x approaches 1 from the left, f(x) approaches 0 from above.
• As x approaches 1 from the right, f(x) approaches 0 from above.

This behavior indicates that the root at x = 1 has a multiplicity of 2, just like the root at x = 0 for f(x) = x².

The graph of f(x) = (x-1)² is essentially the graph of f(x) = x² shifted one unit to the right. This shift does not affect the multiplicity of the root or the general shape of the graph near the root; it simply moves the location of the root from x = 0 to x = 1.

Table of Calculations Close to x=1 for (x-1)²

Behavior Close to x=1 for (x-1)⁴

The function f(x) = (x-1)⁴ behaves in a quartic manner around x = 1, similar to how f(x) = x⁴ behaves around x = 0.

• As x approaches 1 from the left, f(x) approaches 0 from above.
• As x approaches 1 from the right, f(x) approaches 0 from above.

This behavior indicates that the root at x = 1 has a multiplicity of 4, just like the root at x = 0 for f(x) = x⁴.

The graph of f(x) = (x-1)⁴ is essentially the graph of f(x) = x⁴ shifted one unit to the right. This shift does not affect the multiplicity of the root or the general shape of the graph near the root; it simply moves the location of the root from x = 0 to x = 1.

Table of Calculations Close to x=1 for (x-1)⁴

Understanding the Nature of Roots in (x-a)ⁿ for Even n

When dealing with a function like f(x) = (x-a)ⁿ where n is an even integer, it’s important to understand what the exponent means in terms of roots.

Case of (x-a)²

For example, in the case of f(x) = (x-a)², this can be expanded as:

(x-a)² = (x-a)(x-a)

This shows that x = a is a root that appears twice, making it a “double root.”

General Case of (x-a)ⁿ

In the general case of f(x) = (x-a)ⁿ, the root x = a will appear n times. This is known as a root with “multiplicity” of n.

As x approaches a from both the left and the right, f(x) approaches 0 from above. This behavior indicates that the root at x = a has a multiplicity of n.

When you’re very close to the root, a higher n will make the function appear flatter near the root. The graph will approach the x-axis more gradually.

For different values of a such as -1, 0, 1.5, etc., the graph will simply shift horizontally but retain the same general shape and behavior around the root.