Problem 33 - 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. $$y_{1}=\frac{x^{2}}{x+2}, \quad y_{2}=x-2+\frac{4}{x+2}$$

Step-by-Step Solution

Answer

Yes, the two expressions are equivalent as proved by their identical graphs and by algebraic verification.

Step 1: Write down the Equations

First, write down the expressions given: \(y_{1}=\frac{x^{2}}{x+2}\) and \(y_{2}=x-2+\frac{4}{x+2}\).

Step 2: Graph the Two Equations

Now, using a graphing tool - could be online or a physical graphing calculator, graph the two functions \(y_{1}\) and \(y_{2}\) in the same viewing window. Observe that if the two expressions are equivalent, their graphs should coincide. In other words, you should only see essentially one graph.

Step 3: Verify the Equations Algebraically

Now, let's see if the two expressions are indeed equivalent algebraically. Start by simplifying \(y_{2}\) expression which is \(x - 2 + \frac{4}{x+2}\). Multiply every term by \(x+2\) to clear the denominator. You will get \(x(x+2) - 2(x+2) + 4\). Distribute and simplify to finally obtain \(\frac{x^2}{x+2}\), which is the same as \(y_{1}\). The algebraic verification confirms our finding on the graph.

Key Concepts

Equivalent Expressions

Equivalent expressions are vital in mathematics. They simplify complex problems by showing that two different-looking expressions have the same value. In the given exercise, we have two functions:

• \(y_{1} = \frac{x^{2}}{x+2}\)
• \(y_{2} = x - 2 + \frac{4}{x+2}\)

On the surface, these expressions look different. However, by manipulating each side using algebra, we can prove their equality. This involves simplifying the expressions to reveal their underlying equivalence. Understanding this concept helps build a strong foundation for further mathematical operations.

Graphical Verification

Graphical verification is a visual method for proving the equivalence of mathematical expressions. To perform this, use a graphing utility to plot the functions of the two expressions on the same coordinate plane.
To graphically verify:

• Plot \(y_{1} = \frac{x^{2}}{x+2}\)
• Plot \(y_{2} = x - 2 + \frac{4}{x+2}\)

If the graphs overlap completely, it confirms the expressions are equivalent. Graphical verification quickly shows equivalence without intricate calculations. It's beneficial for visual learners who understand concepts better by seeing rather than just calculating.

Algebraic Simplification

Algebraic simplification is an essential skill. It involves reducing expressions to their simplest form. To verify equivalence algebraically, let's simplify the expression \(y_{2} = x - 2 + \frac{4}{x+2}\).
Multiplying every term by \(x+2\) eliminates the faction:

• \(x(x+2) - 2(x+2) + 4\)
• Simplifying gives us: \(x^2 - 2x + 4\)

Continuing with the simplification: Combine like terms to result in the simplest form, which is \(\frac{x^2}{x+2}\), identical to \(y_{1}\). This process proves that even complex terms can be made straightforward through careful manipulation.

Graphing Calculators

Graphing calculators are a powerful tool in mathematics education. They allow students to input expressions, and the calculator will quickly graph them on a display. This feature aids in checking problems involving equivalent expressions.
For instance, to identify the equivalency of \(y_{1}\) and \(y_{2}\):

• Enter both equations into the calculator.
• Observe if the graphs coincide, appearing as a single graph.

Graphing calculators also facilitate exploration of other complex functions beyond manual plotting capabilities, providing practical examples of abstract concepts and making them more accessible.