Problem 60 - Algebra and Trigonometry

Question

Equation Notation on Graphing Calculators When you enter the following equations into your calculator, how does what you see on the screen differ from the usual way of writing the equations? (Check your user's manual if you're not sure.) \(\begin{array}{ll}{\text { (a) } y=|x|} & {\text { (b) } y=\sqrt[5]{x}} \\\ {\text { (c) } y=\frac{x}{x-1}} & {\text { (d) } y=x^{3}+\sqrt[3]{x+2}}\end{array}\)

Step-by-Step Solution

Answer

Calculators use 'abs(x)' for \(|x|\), 'x^{(1/5)}' for \(\sqrt[5]{x}\), require parentheses for fractions, and '{(x+2)}^{1/3}' for cube roots.

Step 1: Absolute Value Function

The standard way to write the absolute value function is \( y = |x| \). On most graphing calculators, you enter this equation as \( y = \text{abs}(x) \) because calculators use the 'abs' function to represent absolute values. The notation '|' for absolute value is typically not available.

Step 2: Fifth Root Function

Standard notation for the fifth root of \( x \) is \( y = \sqrt[5]{x} \). On most calculators, you use an exponent to denote roots. So you would enter it as \( y = x^{(1/5)} \). Calculators typically do not have a specific button for cube or fifth roots, so you use fractional powers instead.

Step 3: Rational Function

The function \( y = \frac{x}{x-1} \) is a rational function. Calculators usually need parentheses to handle divisions properly and avoid ambiguity. Enter this as \( y = (x)/(x-1) \) to ensure the entire expression is calculated as intended. Parentheses are necessary around both the numerator and the denominator.

Step 4: Combination of Polynomial and Cube Root Function

This function mixes polynomial and root operations: \( y = x^{3} + \sqrt[3]{x+2} \). On a calculator, you enter it as \( y = x^{3} + (x+2)^{(1/3)} \). This uses the fractional exponent \( 1/3 \) for the cube root \( \sqrt[3]{x+2} \), and parentheses around \( x+2 \) to ensure the proper root calculation.

Key Concepts

Absolute Value Function

The absolute value function is a unique type of mathematical function that displays the distance of a number from zero on a number line. This means the output, or y-value, is always non-negative, regardless of whether the input, x, is positive or negative. When you see the function written as \( y = |x| \), it's telling you to ignore the sign of \( x \) and only consider its magnitude.
This function plays a significant role in mathematics because it helps simplify equations and inequalities, especially when solving distance-related problems.

• In standard notation: \( y = |x| \)
• On a calculator: \( y = \text{abs}(x) \)

Graphing calculators have a specific way to handle absolute values, often using a built-in function called 'abs'.
By understanding how to input it, you can easily visualize this function's V-shaped graph, which dips at the origin and extends indefinitely. This makes it easy to predict outcomes and solve real-world problems using graphs.

Rational Function

Rational functions involve ratios and hence require careful input to avoid calculation errors, especially on calculators. The function \( y = \frac{x}{x-1} \) is a simple rational function with an undefined value at \( x = 1 \), due to division by zero.
This aspect is key when inputting functions into a graphing calculator.

• The standard way to write it: \( y = \frac{x}{x-1} \)
• On a calculator: \( y = (x)/(x-1) \)

Using parentheses around both the numerator and the denominator ensures that the calculator interprets the function as intended.
When graphed, a rational function can exhibit interesting features like asymptotes or points where the graph approaches but never touches a line. These features reflect crucial changes in the function's behavior at those points.

Fifth Root

Finding roots, such as the fifth root, can be a bit tricky on calculators as they typically lack specific buttons for higher roots. The fifth root, represented as \( y = \sqrt[5]{x} \), simplifies the value by determining what number, when raised to the fifth power, equals \( x \).
For calculators, using fractional exponents is the way to go. This method transforms roots into exponents, making them easier to handle computationally.

• Standard notation: \( y = \sqrt[5]{x} \)
• On a calculator: \( y = x^{(1/5)} \)

Entering \( x \) with an exponent of \( 1/5 \), you tell the calculator to perform the operation for a fifth root.
Graphically, the curve of a fifth root function will increase at a decreasing rate, starting from zero at the origin.

Polynomial and Cube Root Functions

Polynomial functions, such as cubic functions, and their combination with root functions, present their own sets of challenges and opportunities for exploration. The function \( y = x^3 + \sqrt[3]{x+2} \) mixes a polynomial and a cube root, offering a complex yet intriguing graph to analyze.
In calculators, roots are expressed via fractional exponents much like the fifth root.
This specific function is entered as:

• Standard notation: \( y = x^{3} + \sqrt[3]{x+2} \)
• On a calculator: \( y = x^{3} + (x+2)^{(1/3)} \)

The polynomial part elevates \( x \) to the third power, while the cube root transforms \( x+2 \) using a \( 1/3 \) exponent.
This combination results in distinctive graph shapes, showing how polynomial growth at one end interacts with cube-root slows on the other. Understanding how to input and interpret these kinds of functions broadens your ability to solve complex equations and appreciate their graphical beauty.