Problem 34 - 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. $$y_{1}=\frac{x^{2}+2 x-1}{x+3}, \quad y_{2}=x-1+\frac{2}{x+3}$$

Step-by-Step Solution

Answer

The graphs of the two functions are identical, visually proving their equivalence. Algebraically proving \(y_{1}=y_{2}\) confirms their equivalence - therefore, the expressions \(y_{1}=\frac{x^{2}+2 x-1}{x+3}\) and \(y_{2}=x-1+\frac{2}{x+3}\) are equal.

Step 1: Graphing the Equations

First, take the two equations \(y_{1}=\frac{x^{2}+2 x-1}{x+3}\) and \(y_{2}=x-1+\frac{2}{x+3}\) and input them into a graphing utility. Ensure they are in the same viewing window.

Step 2: Visual Verification of Equivalence

Analyze the two graphs. If the expressions \(y_{1}\) and \(y_{2}\) are equivalent, the two graphs should be identical.

Step 3: Algebraic Verification

Now, verify the results algebraically. Rewrite \(y_{2}\) to \(y_{2}=\frac{(x-1)(x+3)+2}{x+3}\). Simplifying gives \(y_{2}=\frac{x^2+2x-1}{x+3}\). So, \(y_{1}=y_{2}\).

Key Concepts

Equivalent Expressions

Equivalent expressions are different algebraic forms that represent the same value for all variables involved. This means that no matter what value is used for any variable within these expressions, their outputs will always be the same. In our exercise, we have the expressions \( y_{1}=\frac{x^{2}+2 x-1}{x+3} \) and \( y_{2}=x-1+\frac{2}{x+3} \). Although these expressions appear different at first glance, they are equivalent, as their algebraic forms can be manipulated and simplified to match one another. Recognizing equivalent expressions is crucial in algebra because it allows for more flexibility in solving equations and understanding mathematical relationships. Additionally, it can simplify complex problems by transforming expressions into more manageable forms.

Algebraic Verification

Algebraic verification involves manipulating expressions through algebraic operations to show that two different-looking expressions are indeed equivalent. In this case, you start with the expressions provided: \( y_{1}=\frac{x^{2}+2 x-1}{x+3} \) and \( y_{2}=x-1+\frac{2}{x+3} \).

To verify algebraically, we modify \( y_{2} \) so it matches \( y_{1} \). We can expand \( y_{2} \) to \( y_{2}=\frac{(x-1)(x+3)+2}{x+3} \). Expanding \((x-1)(x+3)\) gives \(x^2 + 2x - 3\), and adding 2 results in \(x^2 + 2x - 1\). This proves algebraically that both expressions result in \( \frac{x^2+2x-1}{x+3} \).

• Simplifying algebraic expressions allows us to see the underlying equivalence between two forms.
• Follow the steps systematically to correct any computational slips.
• Touch base with each equation part-by-part to ensure they align perfectly in their simplified form.

Algebraic verification provides the 'proof' needed to ensure there are no errors in logical reasoning.

Graphical Verification

Graphical verification is the process of using graphs to assure that two algebraic expressions are equivalent by showing that their graphs are identical. In the given scenario, both \( y_{1}=\frac{x^{2}+2 x-1}{x+3} \) and \( y_{2}=x-1+\frac{2}{x+3} \) are entered into a graphing utility. If the expressions are equivalent, their graphs will be identical across their domains.

Here's how to approach it:

• Input both equations into the graphing utility.
• Adjust the viewing window to encompass key points, such as intercepts and asymptotes, visibly.
• Examine the appearance of each graph to check for superposition or alignment. They should superimpose completely if they represent equivalent expressions.

In doing so, graphical verification offers a visual confirmation of equivalence, which supports the algebraic findings and provides a more intuitive understanding of the relationships between expressions.