Problem 49 - Algebra for College Students

Question

How should the graphs of \(f(x)=x^{3}, f(x)=x^{5}\), and \(f(x)=x^{7}\) compare? Graph these three functions on the same set of axes.

Step-by-Step Solution

Answer

All three graphs intersect at the origin and exhibit an S-shape, but higher powers rise and fall more steeply away from the origin.

Step 1: Understand the Functions

Each of these functions are power functions, where the exponent is an odd integer: 3, 5, and 7 respectively. These functions are of the form \( f(x) = x^n \) and they all pass through the origin (0,0) because any number raised to any power is zero at this point.

Step 2: Determine Basic Shape

For odd powers, the graph has a specific S-shape, which is symmetric about the origin. This means these functions are odd functions. As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.

Step 3: Compare Rates of Increase

Higher degree polynomials grow faster as \( x \) moves further from zero. Thus, \( f(x) = x^7 \) increases more steeply than \( f(x) = x^5 \), which in turn increases more steeply than \( f(x) = x^3 \) as \( x \) moves away from zero. Near the origin, \( f(x) = x^3 \) appears closer to the diagonal, while \( f(x) = x^7 \) and \( f(x) = x^5 \) are flatter.

Step 4: Intersection at the Origin

Since all three functions pass through the origin, they all intersect at the point \( (0,0) \). This is a crucial point where the graphs meet.

Step 5: Sketch the Graphs

Plotting these curves, you will see that all three functions pass through the origin and exhibit an increasing S-shape. \( f(x) = x^3 \) will be the least steep; \( f(x) = x^5 \) will be steeper; and \( f(x) = x^7 \) will be the steepest away from the origin.

Key Concepts

Odd Functions

Odd functions, like those explored in the exercise, have a unique property called rotational symmetry about the origin.
For a function to be classified as odd, the equation must satisfy this: \[ f(-x) = -f(x) \] This means if you rotate the graph 180 degrees around the origin, it looks the same.

Examples from the exercise include functions like \( f(x) = x^3 \), \( f(x) = x^5 \), and \( f(x) = x^7 \), all of which are power functions with odd exponents. Being odd functions means that as you graph them, they will always intersect the origin and have lines (or curves) that mirror each other quadrantly.

Power Functions

Power functions follow the general form \( f(x) = x^n \), where \( n \) is a real number. In this exercise, we predominantly focus on odd integer exponents, which directly affect the graph's shape.

Characteristics of these functions include:

• For odd exponents (like 3, 5, and 7), the graph typically forms an "S" shape, indicating a steep increase or decrease dependent on the sign of \( x \).
• Since the power is the highest degree in the polynomial, it's the primary determinant of how rapidly the function grows or declines as \( x \) moves away from zero.

It's useful to remember that power functions with higher exponents will have steeper curves.

Graphing Functions

Graphing functions, such as in the exercise, link visual representation with algebraic expression.
When plotting functions like \( f(x) = x^3 \), \( f(x) = x^5 \), and \( f(x) = x^7 \), you'll notice:

• All graphs pass through the origin (0,0), a common feature of power functions.
• The graph displays a twisted path due to the odd nature of these functions, creating P or S-shaped curves dependent on orientation.

These graphs show how varying the power function's exponent changes its slope and curvature, making graphing an invaluable tool in visualizing and understanding function behavior.

Rates of Increase

In analyzing rates of increase, particularly for polynomials with odd exponents, the higher the exponent, the steeper the graph.

• For instance, \( f(x) = x^7 \) escalates at a faster rate than \( f(x) = x^5 \), which in turn is steeper than \( f(x) = x^3 \).
• Such differences are most evident as \( x \) values move away from the origin, where larger exponents dominate the rate of increase or decrease.

Near the origin, the functions are relatively flat, emphasizing the initial influence of lower degrees before higher power terms begin to take over the graph's steepness.