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Magazine Chapter 5: Problem 44
For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely with \(x\) and when \(x=6, y=2\).
Short Answer
Expert verified
Graph the equation \( y = \frac{12}{x} \).
Step by step solution
01
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: \[ y = \frac{k}{x} \].
02
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: \( k = 12 \).
03
Writing the Specific Equation
Now that we have \( k = 12 \), the specific inverse variation equation becomes: \[ y = \frac{12}{x} \].
04
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 \( x \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \( y \).
- 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 \( k \).
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 \( x \).
- As \( x \) grows larger, \( y \) gets smaller.
- As \( x \) decreases, \( y \) grows larger.
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.
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:
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.
- 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.
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.
