Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Comparing Graphs Use a graphing utility to graph the functions given by \(f(x)=x^{3}, g(x)=x^{5}\), and \(h(x)=x^{7} .\) Do the three functions have a common shape? Are their graphs identical? Why or why not?

Step-by-Step Solution

Answer

No, the graphs of the three functions are not identical. They exhibit a common point at the origin and show general similarity as odd functions, but they differ in their rates of change with the increase or decrease of x-values.

Step 1: Plot the Functions

Use a graphing utility to plot the functions \(f(x)=x^{3}, g(x)=x^{5}\), and \(h(x)=x^{7}\). Observe the shapes of their graphs, their direction, their maximum and minimum points if any.

Step 2: Compare the Shape

Compare the three graphs in terms of their general shape. At first glance, you might observe that all three functions have a similar shape. They are all odd functions with positive and negative values for positive and negative x-values and a common point at the origin (0,0).

Step 3: Compare the Specific Characteristics

Upon closer inspection, notice that while the general shape is similar, the graphs are not identical. For smaller values of x (both negative and positive), the rate of change of \(f(x)=x^{3}\) is slower than that of \(g(x)=x^{5}\) and \(h(x)=x^{7}\). As x tends to large positive or large negative values, the shapes of \(g(x)=x^{5}\) and \(h(x)=x^{7}\) becomes steeper compared to \(f(x)=x^{3}\). This shows that the graphs are not identical.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that consist of multiple terms. These terms are typically composed of a variable (like 'x') raised to an integer power and multiplied by a coefficient. The functions in our exercise, such as \(f(x)=x^3\), \(g(x)=x^5\), and \(h(x)=x^7\), are examples of polynomial functions. These functions are called monomials because they have only one term each.Polynomial functions can have differing degrees, which is determined by the highest power of the variable in the expression. For instance, \(f(x)=x^3\) is a polynomial function of degree 3. A function's degree offers insights into its graph's behavior, such as the number of roots or intersections it can have with the x-axis, as well as the end behavior at positive or negative infinity.Understanding polynomial functions is integral to analyzing their graph shapes and predicting their behavior across various values of x. In the context of our example, the increase in degree from 3 to 5 to 7 changes the steepness and curvature of the graph significantly.

Graphical Analysis

Graphical analysis involves evaluating the visual representation of functions on a coordinate plane. With functions like \(f(x)=x^3\), \(g(x)=x^5\), and \(h(x)=x^7\), graphical analysis can help visualize and compare their behaviors in a more intuitive way.By plotting these functions, one can observe important features. For instance:

• All three functions intersect at the origin (0,0), which is a common point shared by odd-powered polynomials.
• The symmetry observed around the origin is a characteristic of odd functions.
• The steeper the graph of a function, the greater the rate of change for that function as the values of x increase or decrease.

Graphical analysis allows us to identify such characteristics by visually dissecting key components on the graph. This not only helps in understanding the shape and direction of functions but also in comparing their distinct and shared properties.

Function Comparison

Function comparison involves examining the differences and similarities between multiple mathematical functions. When comparing \(f(x)=x^3\), \(g(x)=x^5\), and \(h(x)=x^7\), it's important to note both their obvious similarities and subtle differences.

• Common Shape: While all three functions display a similar basic shape due to their odd degrees, they each exhibit unique characteristics as a result of their varying degrees. All share a point of intersection at the origin.
• Distinct Steepness: One key difference is how steeply they rise or fall. The higher degree functions, \(g(x)=x^5\) and \(h(x)=x^7\), show more rapid increases or decreases compared to \(f(x)=x^3\) for the same x-values. This is because the polynomial's leading degree influences the graph's steepness.
• Asymptotic Behavior: As x moves towards positive or negative infinity, each function’s curve becomes more exaggerated, pointing more steeply upwards or downwards depending on the coefficient's sign. It’s valuable to appreciate these differences when performing graphical comparisons or making predictions about function behavior under various scenarios.

Understanding these comparative aspects allows learners to develop a deeper insight into polynomial functions and effectively leverage their distinctive properties.