Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the six functions below in the same viewing window. Describe any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

Step-by-Step Solution

Answer

All functions pass through the origin and increase when \(x>0\). Even-powered functions increase and become flatter near the origin as the power increases when \(x<0\), while odd-powered functions decrease and retain a linear shape resembling \(y=x\) when \(x<0\).

Step 1: Graphing the functions

Using a graphing utility, graph the six functions \(y=x\), \(y=x^{2}\), \(y=x^{3}\), \(y=x^{4}\), \(y=x^{5}\), and \(y=x^{6}\) individually.

Step 2: Observe similarities and differences

Analyzing the graphs, we notice that all functions are increasing and pass through the origin, \(0,0\), showing a similarity between all functions. When \(x>0\), all functions are also increasing which is another common feature. However, the rate at which they increase varies and this is where they start to differ. When \(x<0\), the functions \(y=x\), \(y=x^{3}\), and \(y=x^{5}\) decrease, while the others \(y=x^2, y=x^4 \), and \(y=x^6\) increase, demonstrating a key difference between them. The shapes of functions become increasingly flat near the origin for even powers, whereas for odd powers, they resemble more the linearity of the function \(y=x\).

Step 3: Conclude the properties and behavior of the functions

From the plotted graphs and observed similarities and differences, it is concluded that all functions pass through the origin, \(0,0\), and are increasing when \(x>0\). The differences lie in the behavior when \(x<0\) where even-powered functions increase and odd-powered functions decrease. Also, as the power increases, the function graph flattens near the origin for even powers, while odd powers remain linearly more similar to \(y=x\).

Key Concepts

Graphing Functions

When graphing functions, especially polynomial functions like those in the exercise, it's crucial to understand how the general shape of the graph changes with different exponents. Graphing involves plotting a function on a coordinate system to visualize its shape, behavior, and other characteristics. All these functions pass through the origin. This means the point (0,0) is a common point where the graphs intersect the axes. You might use a graphing utility to make this process easier, especially for higher-degree polynomials which can become complex.

• Linear function (y = x) : This is a straight line passing through the origin at a 45-degree angle to the axes.
• Quadratic function (y = x^2) : A U-shaped parabola opening upwards, symmetric about the y-axis.
• Cubic function (y = x^3) : An S-shaped curve that traverses through the origin, displaying symmetry about the origin itself.
• Quartic function (y = x^4) : Similar to a quadratic but wider at the base and flatter near the origin.
• Quintic function (y = x^5) : Resembles the cubic function but with steeper slopes as it moves away from the origin.
• Sextic function (y = x^6) : Wider and flatter near the origin like the quartic, but more pronounced.

Each graph reveals how polynomial degree affects the curve's overall behavior and complexity.

Even and Odd Functions

Even and odd functions have distinct characteristics that are important to understand when analyzing graphs. This is especially true for polynomial functions. An even function has a symmetry about the y-axis, while an odd function is symmetric about the origin.

• Even Functions ( (y = x^2), (y = x^4), (y = x^6) ): These functions rise for both positive and negative values of x. Their symmetry means that if you fold the graph along the y-axis, the two halves would overlap perfectly.
• Odd Functions ( (y = x), (y = x^3), (y = x^5) ): These functions, when reflected across both axes, would match their original shape. They decrease when x is negative and increase when x is positive, maintaining symmetry about the origin.

This pattern influences how the graph stretches or flattens. Even functions tend to be flatter at the origin compared to their odd counterparts due to their degree.

Function Behavior

The behavior of functions, particularly polynomial functions, reveals itself as you observe how these equations progress based on their degree. The degree determines how fast the function rises or falls and affects the graph's shape and approach to infinity.

• When (x > 0) : All functions increase. However, as the degree increases, the function rises more steeply. For instance, (y = x^6) rises much faster than (y = x^2) for the same positive x value.
• When (x < 0) : This behavior diverges between even and odd functions. Even functions like (y = x^2) still rise, due to the squaring of negative numbers resulting in positives. In contrast, odd functions, e.g., (y = x^3) , decrease as negative inputs result in negative outputs.
• Near the origin: The behavior is also distinctive. Even functions flatten while odd functions remain more linear in appearance similar to (y = x) .

Understanding these behaviors helps to predict the shape and other qualitative aspects of the function's graph. It also makes it easier to anticipate how modifications to the function will alter its graphation.