Problem 77 - Algebra and Trigonometry

Question

Enter Equations Carefully A student wishes to graph the equations $$ y=x^{1 / 3} \quad \text { and } \quad y=\frac{x}{x+4} $$ on the same screen, so he enters the following information into his calculator: $$ Y_{1}=x^{\wedge} 1 / 3 \quad Y_{2}=x / x+4 $$ The calculator graphs two lines instead of the equations he wanted. What went wrong?

Step-by-Step Solution

Answer

The student needed to use parentheses: \( Y_1 = x^{(1/3)} \) and \( Y_2 = \frac{x}{(x+4)} \).

Step 1: Analyze Y1 equation

The student entered the equation for \( Y_1 \) as \( x^{\wedge} 1 / 3 \). However, calculators interpret exponentiation as \( a^{b} / c \) if not properly grouped. Thus, the calculator computes \( x \) raised to the power of \( 1 \), then divided by \( 3 \), effectively graphed as \( y = \frac{x}{3} \). This results in a line, not the intended curve \( y = x^{1/3} \).

Step 2: Analyze Y2 equation

The student entered the equation for \( Y_2 \) as \( x / x+4 \). Without proper brackets, the calculator follows the order of operations and reads \( \frac{x}{x} + 4 \) instead, which simplifies to \( 1 + 4 = 5 \) for all non-zero \( x \). This plots as the horizontal line \( y = 5 \), not the intended function \( y = \frac{x}{x+4} \).

Step 3: Correct Y1 equation entry

To correctly graph \( y = x^{1/3} \), the student should use parentheses to dictate grouping: \( Y_1 = x^{(1/3)} \). This ensures the calculation is interpreted as \( x \) raised to the power of \( 1/3 \), not divided by 3.

Step 4: Correct Y2 equation entry

For the second equation \( y = \frac{x}{x+4} \), the expression must be entered with parentheses: \( Y_2 = \frac{x}{(x+4)} \). This ensures the denominator is correctly recognized as \( x + 4 \) rather than just \( x \).

Step 5: Test and Verify

After correcting the entries, the student should re-enter the equations into the calculator as \( Y_1 = x^{(1/3)} \) and \( Y_2 = \frac{x}{(x+4)} \). Graph the functions and verify that the intended curve and rational function are displayed, not linear equations.

Key Concepts

Calculator Entry Errors

When entering equations into a calculator, especially complex ones, it's easy to make mistakes. The most common error is incorrect entry. Calculators require precise input because they interpret equations very literally.

For instance, when you enter an expression without necessary symbols like parentheses, the calculator may misinterpret the equation. In our example, the calculator misread the equations, which led to unexpected graphs like lines instead of curves or nonlinear graphs.

Such errors are avoidable when you’re aware of how the calculator processes mathematical operations. Always double-check your entries before expecting accurate outputs.

Exponentiation and Division

Calculators process exponentiation and division by following strict rules. Exponentiation is raising a number to the power of another. Division splits one number by another.

In the equation \(y = x^{1 / 3}\), the student intended the cube root of \(x\). However, the input \(x^{\wedge} 1 / 3\) suggests dividing after exponentiation, resulting in \(y = \frac{x}{3}\), not the desired graph.

• Use parentheses for exponents: \(x^{(1/3)}\) ensures correct processing.
• Verify exponent placement to avoid unintentional divisions.

Order of Operations

The order of operations is crucial when dealing with calculators. This is the sequence in which different operations are processed:

• Parentheses first
• Exponents (powers and roots)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Misunderstanding or skipping this order can yield drastic errors.

In our example, \(x / x+4\) was interpreted by the calculator as \(\frac{x}{x} + 4\), due to its order-evaluating rules, which resulted in a consistent \(y = 5\) for non-zero \(x\). Knowing the order helps avoid mistakes.

Parentheses in Expressions

Parentheses are vital in directing the flow of calculations. They specify which parts of an equation should be computed first.

Let's revisit the equation \(x / x+4\), which the user meant to be viewed as \(\frac{x}{x+4}\). By explicitly surrounding \((x+4)\) with parentheses, the calculator sees it as a unit.

• Parentheses change how equations are interpreted and solved.
• Using them properly ensures correct placement of terms within complex expressions.

Double-check the grouping in expressions before entering them into calculators.