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Chapter 3: Problem 12

Display the graphs of the given functions on a graphing calculator. Use appropriate window settings. $$y=x^{2}-3 x^{5}+3$$

Short Answer

Expert verified
Use a window with \( x = [-3, 3] \) and \( y = [-10, 10] \) to graph \( y = x^2 - 3x^5 + 3 \). Adjust if needed.

Step by step solution

01

Identify the Function

We are given the function \( y = x^2 - 3x^5 + 3 \). This function is a polynomial of degree 5.
02

Analyze the Function

Since \( y = x^2 - 3x^5 + 3 \) includes terms like \( x^2 \) and \( -3x^5 \), this is a polynomial function that tends to \( -\infty \) as \( x \to \pm\infty \) due to the negative coefficient of the highest degree term.
03

Determine the Domain and Range

For a polynomial function, the domain is all real numbers \(( -\infty, \infty )\). The range can be evaluated by testing endpoints and considering the symmetry/pattern. The actual window will help find the range more accurately.
04

Choose Appropriate Window Settings

To effectively display the function behavior, select a window setting like \( x \) from \( -3 \) to \( 3 \) and \( y \) from \( -10 \) to \( 10 \). Adjust if the curve isn't fully captured.
05

Plot on Graphing Calculator

Enter the function \( y = x^2 - 3x^5 + 3 \) into the calculator. Set the window as previously described and display the graph.
06

Analyze the Graph

Examine the shapes and intercepts. The graph will show turning points and roots; note particularly any maxima or minima and where the graph crosses the axes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polynomial Degree
The degree of a polynomial is critical in determining its behavior and number of roots. For the function \( y = x^2 - 3x^5 + 3 \), the highest power of \( x \) is 5, thus this is a fifth-degree polynomial. The degree tells us several important things:
  • It indicates the maximum number of roots or x-intercepts the polynomial can have. For a fifth-degree polynomial, there can be up to 5 real roots.
  • The degree also affects the end behavior of the graph. Here, because the leading term \(-3x^5\) has a negative coefficient, as \( x \to \infty \) or \( x \to -\infty \), \( y \) will also tend towards \(-\infty\).
This helps us predict how the graph extends beyond the visible section on the graphing calculator. Every polynomial degree offers clues about the shape and complexity of the graph.
Domain and Range of Polynomials
For any polynomial function like \( y = x^2 - 3x^5 + 3 \), the domain is all real numbers \( (-\infty, \infty) \). This is because you can substitute any real number for \( x \) without restrictions. The range, which is the set of all possible \( y \)-values, can be trickier to determine.
The range depends on:
  • The degree and coefficients of the polynomial, which influence the direction and extent of the graph.
  • The behavior of the graph at "endpoints" for very large or small \( x \) values, which often helps understand the boundaries of \( y \).
Utilizing a graphing calculator to visually assess the range is beneficial. As you adjust window settings, you can better estimate the full set of outputs, or \( y \)-values, the function will attain.
Mastering Graphing Calculator Usage
A graphing calculator is a powerful tool for visualizing polynomial functions. To graph \( y = x^2 - 3x^5 + 3 \), start by entering the function:
  • Turn on your calculator and select the graphing function mode.
  • Input \( y = x^2 - 3x^5 + 3 \) using the calculator's keypad and function keys.
Next, adjust the display settings to ensure the graph fits well on the screen. Understanding how to navigate your calculator鈥檚 interface and input functions accurately helps in accurately analyzing the graph produced. Tool familiarity is key in quickly assessing polynomial behavior and estimates of important graph features.
Using Window Settings Effectively
Selecting the right window settings is crucial to effectively graphing polynomial functions on a calculator. For the function \( y = x^2 - 3x^5 + 3 \), appropriate window settings help in complete visualization.
  • Start by setting \( x \) values from \(-3\) to \(3\), which keeps the view concentrated around the origin where most significant behavior might happen.
  • For the \( y \)-axis, set between \(-10\) to \(10\). This range ensures that peaks and troughs of the graph are visible.
After graphing the polynomial, check if the whole behavior of the function is visible. Adjust the window accordingly if the graph expands outside the preset view. These settings help capture the essence of the polynomial鈥檚 graphical representation, ensuring key features like intercepts and turning points are not missed.

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Most popular questions from this chapter

Solve the given problems. The sales tax in a city is \(7.00 \% .\) The price tag on an item is \(D\) dollars. If the store has a \(19 \%\) off price tag sale, express the total cost of the item as a function of \(D\)

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