Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{4}-3 x^{2}-1}{x^{2}+5}, \quad y_{2}=x^{2}-8+\frac{39}{x^{2}+5}$$

Step-by-Step Solution

Answer

The expressions \(y_{1} = \frac{x^{4}-3 x^{2}-1}{x^{2}+5}\) and \(y_{2} = x^{2}-8+\frac{39}{x^{2}+5}\) are equivalent, as confirmed by both graphing and algebraic verification.

Step 1: Graphing the Functions

By using a graphing utility, plot the functions \(y_{1}=\frac{x^{4}-3 x^{2}-1}{x^{2}+5}\) and \(y_{2}=x^{2}-8+\frac{39}{x^{2}+5}\). Observe the graphs of both functions. If they overlap, this indicates that the functions are identical.

Step 2: Algebraic Verification

Verify the results algebraically by setting \(y_{1}\) equal to \(y_{2}\) and simplifying the equation. Set \(\frac{x^{4}-3 x^{2}-1}{x^{2}+5}\) equal to \(x^{2}-8+\frac{39}{x^{2}+5}\) and simplify. In the first step, multiply each side of the equation by \(x^{2}+5\) to clear the fraction:\(x^{4}-3 x^{2}-1 = (x^{2}-8)(x^{2}+5) + 39\)In the next step, expand \(x^{2}-8\) into \(x^{2}+5\), so the equation becomes:\(x^{4}-3 x^{2}-1 = x^{4}-8x^{2}+5x^{2}-40+ 39\)This simplifies to:\(x^{4}-3 x^{2}-1 = x^{4}-3x^{2}-1\)This verifies that \(y_{1}\) is indeed equivalent to \(y_{2}\).

Key Concepts

Graphing Utility

To graph functions effectively, one can employ a graphing utility. This tool simplifies the task of plotting complicated equations by automating the graphing process. A graphing utility refers to either a calculator or a software tool that can visualize the relationship between variables in an equation.

Graphing utilities often provide interactive features, allowing users to zoom in on specific parts of the graph or change the viewing window. This versatility helps with observing whether two graphs coincide, hinting that the functions represented by the equations are equivalent. Using a graphing utility streamlines the process of comparing functions, making it easier to spot similarities such as overlapping curves.

• They save time and reduce manual calculation errors.
• Help in educational settings by enhancing understanding of graph behaviors.
• Makes graph comparison much easier and accurate.

Equivalent Expressions

In mathematics, equivalent expressions are expressions that, although they may look different, represent the same value for any value of the variables involved in them. Two expressions are equivalent if they yield the same result when evaluated.
For example, in the provided exercise, the expressions for \(y_1\) and \(y_2\) appear different, yet both equations define the same function. Detecting equivalent expressions can simplify mathematical computations and deepen understanding of function behaviors.

• Be sure to verify equivalence both graphically and algebraically.
• It is crucial in simplifying complex problems by recognizing shared characteristics.

Understanding equivalence helps immensely in mathematics as it aids in solving, simplifying, and manipulating equations in an efficient manner.

Simplification

Simplification is a mathematical process that involves reducing an expression or equation to its simplest form. This is done by performing algebraic operations to condense the expression, while still preserving its equivalence.
In this context, simplification involves clearing fractions, expanding terms, and combining like terms to verify that two expressions are indeed equivalent.
When attempting to prove that \(y_1 = y_2\), the expressions are simplified step-by-step until both sides match exactly. Doing so confirmed the equality of the two functions:

• Simplification steps help in validating algebraic equivalence.
• It is a foundational skill that allows for clearer analysis of complex equations.
• Simplified forms are easier to work with and reduce errors during calculations.

Mastering simplification is vital as it underpins more advanced mathematical operations and proofs.

Graphing Functions

Graphing functions is a crucial method in understanding how an equation behaves visually. By representing a function with a graph, one can easily examine the relationship between the input variable \(x\) and the output variable \(y\).
Graphing helps in distinguishing characteristics like intercepts, slope, and curvature. In the context of the exercise, graphing both equations \(y_1\) and \(y_2\) gives a visual confirmation of their equivalence since their graphs overlap completely. This overlap visually demonstrates that both functions, despite their different algebraic forms, share the same set of points in the viewing window.

• Enables easy visualization of the function's characteristics.
• Facilitates comparison between different functions.
• Provides immediate visual proof of equivalence when overlaps occur.

Graphing is an indispensable skill that complements algebraic methods by providing a visual context to abstract mathematical concepts.