Problem 52 - Precalculus

Question

(a) Graph the equation \(y=20 / x\) using a standard viewing rectangle. (b) Although both the \(x\) - and the \(y\) -axes are asymptotes for this curve, the graph in part (a) does not show this clearly. Take a second look, using a viewing rectangle that extends from -100 to 100 in both the \(x\) -and the \(y\) -directions. Note that the curve indeed appears indistinguishable from an asymptote when either \(|x|\) or \(|y|\) is sufficiently large.

Step-by-Step Solution

Answer

The graph's asymptotic nature is clearer in a wider viewing range from -100 to 100.

Step 1: Understanding the Equation

The given equation is \( y = \frac{20}{x} \), which is a hyperbola. We will analyze its basic properties and identify its asymptotes.

Step 2: Identifying Asymptotes

For a hyperbola \( y = \frac{k}{x} \), both the x-axis (\( y = 0 \)) and y-axis (\( x = 0 \)) are asymptotes. This equation will approach these lines as \( x \) becomes very large or very small.

Step 3: Graphing in a Standard Rectangle

We'll plot the hyperbola using a standard viewing rectangle, which typically spans a smaller range (e.g., \(-10 \) to \( 10 \) on both axes). In this window, the asymptotic behavior might not be very clear, especially near the axes.

Step 4: Observing Graph in an Expanded Range

To more clearly observe the asymptotic behavior, plot the graph in a viewing window that extends from \(-100\) to \(100\) on both axes. This broader view should make it more apparent how the curve becomes closer to the x-axis and y-axis as \( |x| \) or \( |y| \) grows very large.

Key Concepts

Asymptotes

In mathematical terms, an asymptote is a line that a graph approaches but never touches. When dealing with hyperbolas, such as the equation \( y = \frac{20}{x} \), the asymptotes are crucial for understanding the curve's behavior.

For this particular equation, there are two asymptotes:

• The x-axis, represented by the line \( y = 0 \), which the curve approaches as \( x \) becomes very large or very small.
• The y-axis, represented by the line \( x = 0 \), which the curve nears as \( y \) increases or decreases significantly.

The hyperbola never actually meets these axes, no matter how far you extend your graph. This demonstrates that as the variables increase in absolute value, the curve continues to approach, but will never cross, these lines.

Graphing Techniques

Graphing a hyperbola requires careful attention to detail, particularly when dealing with its asymptotic nature. To properly graph \( y = \frac{20}{x} \), start with plotting key points where the x and y values may show interesting behavior or trends.

A good starting step is to examine this equation for various \( x \)-values:

• When \( x = 1 \), \( y \) becomes \( 20 \).
• When \( x = -1 \), \( y \) is \(-20\).
• Consider \( x \) greater than 1 or less than -1 to see how \( y \) diminishes closer to the x-axis.

The symmetry in hyperbolas aids in easier plotting. If you calculate points in one quadrant, you have an idea of what to expect in the others. Additionally, recalling the asymptotes helps sketch the hyperbola more accurately since they act as a guiding "framework" that informs where the curve is heading.

Viewing Rectangle Adjustment

When graphing functions like hyperbolas, adjusting the viewing rectangle is a crucial step to comprehend the full behavior of the graph. In the context of our hyperbola, the standard viewing rectangle might not adequately display the curve's asymptotic nature.

A typical standard viewing rectangle might cover bounds like \(-10\) to \(10\) on both axes, which could obscure the hyperbola's approach towards its asymptotes. To improve visualization:

• Extend the viewing limits to \(-100\) to \(100\) for both the x and y axes.
• This expansion allows the hyperbola to reveal its pattern of approaching the axes without actually touching them, thus demonstrating the true path along which the curve travels.
• The broader window ensures that the subtlety of asymptotic behavior, either flattening along the x-axis or narrowing towards the y-axis, becomes apparent.

By making this adjustment, students are better able to see how the function behaves for larger values, allowing for a more complete understanding of the hyperbolic relationship.