Question

Use a graphing utility. Graph \(y=x^{3}, y=x^{3},\) and \(y=x^{7}\) on the same screen. What do you notice is the same about each graph? What do you notice is different?

Short answer

All graphs are odd, symmetric about the origin, and pass through \((0,0)\). Higher-degree polynomials are flatter near the origin and steeper away from it.

Step-by-step solution

- Understand the Equations

The equations given are polynomial functions: 1. \( y = x^{3} \) (cubic function) 2. \( y = x^{5} \) (quintic function) 3. \( y = x^{7} \) (septic function). These are all odd-degree polynomials.

- Use a Graphing Utility

Use any graphing utility (like Desmos, GeoGebra, or a graphing calculator) to graph all three equations on the same coordinate plane. Input the equations individually and ensure they are displayed together.

- Observe the Graphs

Observe the shapes and behaviors of the graphs of the equations. Notice how each curve passes through the origin \((0,0)\) and extends towards \((-∞, -∞)\) and \((∞, ∞)\).

- Identify Similarities

All graphs are odd functions, symmetric about the origin, and pass through the origin. They all have similar 'S' like shapes because these are odd-degree polynomials.

- Identify Differences

Higher degree polynomials like \( y = x^{7} \) are flatter near the origin and steeper as \( x \) moves away from 0, compared to \( y = x^{3} \) and \( y = x^{5} \). As the degree increases, the graph becomes steeper faster.

Key concepts

Understanding Odd-Degree Polynomials

Odd-degree polynomials, like those in the given exercise, are functions where the highest power of the variable is an odd number. In this exercise, the functions are cubic ( y = x^3 ), quintic ( y = x^5 ), and septic ( y = x^7 ) polynomials.

These polynomials have distinct characteristics:

• They all pass through the origin (0,0) .
• Their general shape resembles an 'S'-curve.
• The ends of the graph extend towards (- ∞, ∞ ) and ( ∞, ∞ ) as x moves away from zero in both positive and negative directions.

One interesting thing about odd-degree polynomials is that they are symmetric about the origin. This means the graph of an odd-degree polynomial looks the same if rotated 180 degrees about the origin.

The general behavior of these functions is quite uniform hence making predictions about the graph’s appearance more intuitive once you know the degree of the polynomial.

Utilizing a Graphing Utility

To graph polynomial functions effectively, it's highly recommended to use a graphing utility. Tools like Desmos, GeoGebra, or even a graphing calculator can help you visualize these functions easily.

Here’s how you can use a graphing utility:

• Input each of the equations ( y = x^3 , y = x^5 , and y = x^7 ) into the graphing tool.
• Make sure all graphs are displayed on the same coordinate plane to compare them directly.
• Adjust the scale if necessary to observe the shape and behavior of the graphs clearly.

Seeing the graphs on the same plane will help you directly observe their similarities and differences. For example, you'll notice how they all pass through (0,0) and extend in opposite directions towards infinity.
Using these tools not only makes the process easier, but it also provides a visual aid which is crucial in understanding the behavior of polynomials.

Exploring Function Behavior

Understanding the behavior of polynomial functions is key in graph analysis. Odd-degree polynomials, in particular, have unique traits worth noting.

Firstly, let's look at what’s similar:

• All graphs are odd functions meaning they exhibit rotational symmetry about the origin.
• They all intersect the origin (0,0) .
• Each function displays an 'S'-shaped curve.

When it comes to differences:

• Higher degree polynomials like y = x^7 are noticeably flatter near the origin compared to y = x^3 and y = x^5 .
• As you move away from zero, higher-degree polynomials get steeper faster. This means that y = x^7 will rise or fall more sharply compared to y = x^3 or y = x^5 .

The higher the degree of the polynomial, the more extreme these behaviors become. Observing these properties helps in predicting how alterations in polynomial equations affect their graphs, which is crucial for more advanced mathematical studies.
Understanding these behaviors not only aids in academic exercises but also in real-world applications where polynomial functions are used to model various phenomena.