Problem 44 - College Algebra

Question

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely with \(x\) and when \(x=6, y=2\).

Step-by-Step Solution

Answer

Graph the equation \( y = \frac{12}{x} \).

Step 1: Understanding Inverse Variation

If a quantity \( y \) varies inversely with \( x \), it means \( y \) is equal to a constant \( k \) divided by \( x \). The equation that captures this relationship is: \[ y = \frac{k}{x} \].

Step 2: Finding the Constant of Variation

To find the constant \( k \), we use the given point \( (x, y) = (6, 2) \). Substitute these values into the inverse variation equation: \( 2 = \frac{k}{6} \). Solve for \( k \) by multiplying both sides by 6: \( k = 12 \).

Step 3: Writing the Specific Equation

Now that we have \( k = 12 \), the specific inverse variation equation becomes: \[ y = \frac{12}{x} \].

Step 4: Graphing the Equation

Using a calculator, enter the equation \( y = \frac{12}{x} \) to plot the graph. The graph will show a hyperbola, considering both positive and negative values of \( x \).

Step 5: Analyzing the Graph

Observe that as \( x \) increases, \( y \) decreases, and vice versa, reflecting the inverse relationship. The graph should approach but not touch the x-axis or y-axis, serving as asymptotes.

Key Concepts

Constant of Variation

The constant of variation, denoted as \( k \), is essential in understanding inverse variation. Inverse variation describes a relationship where one quantity increases as another one decreases. In simpler terms, if \( y \) varies inversely with \( x \), then \( y = \frac{k}{x} \). Here's how it works:

• We need a pair of values \( (x, y) \) to find \( k \).
• In the given problem, the values \( x = 6 \) and \( y = 2 \) help us find \( k \).

Plug these values into the equation to get \( 2 = \frac{k}{6} \). Solve this equation by multiplying both sides by 6, which results in \( k = 12 \).
Once you know \( k \), you can describe the entire relationship with the specific equation \( y = \frac{12}{x} \). This constant allows you to understand how changes in \( x \) will inversely affect \( y \).

Graphing Equations

Graphing an inverse variation equation, like \( y = \frac{12}{x} \), gives you a visual understanding of how the two variables interact. Employ a calculator to plot this equation, and observe how \( y \) changes as \( x \) changes. Here's what you'll see:

• The graph forms a curve called a hyperbola.
• The curve falls in both the positive and negative quadrants for \( x \).

This graphical representation reveals that:

• As \( x \) grows larger, \( y \) gets smaller.
• As \( x \) decreases, \( y \) grows larger.

Overall, graphing these equations helps you better understand the inverse relationship and allows you to predict how \( y \) will behave as \( x \) changes.

Hyperbola

A hyperbola is the shape formed by the graph of an inverse variation equation such as \( y = \frac{12}{x} \). This shape arises because of the unique inverse relationship between the two variables.Key features of a hyperbola include:

• Two separate curves, one in each of the quadrants with positive and negative values of \( x \).
• The curves never touch the x-axis or y-axis but get infinitely close to them as \( x \) approaches zero.

The hyperbola highlights the difference between direct and inverse variation. Unlike direct variation, where two variables increase or decrease together, a hyperbola shows that while one variable increases, the other decreases. This special structure allows us to understand complex mathematical relationships better.

Asymptotes

Asymptotes are lines that the graph of an equation approaches but never actually touches. In the graph of \( y = \frac{12}{x} \), the x-axis and y-axis act as asymptotes. Understanding asymptotes is crucial because:

• They represent boundaries for the hyperbola on the graph.
• As \( x \) becomes very large or very small (approaching zero), \( y \) will get closer to the axes but will never intersect.

These lines are significant because they suggest what happens as the values become extremely large or extremely small.
This helps predict the behavior of the function and gives insights into the characteristics of inverse variation. Asymptotes are a vital part of understanding how the relationship stretches to infinity and provides a roadmap of sorts for graphing complex equations.