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Magazine Chapter 10: Problem 39
Describe how the graph of \(y^{2}-\frac{x^{2}}{k^{2}}=1\) changes as \(|k|\) increases.
Short Answer
Expert verified
As \(|k|\) increases, the hyperbola becomes narrower along the x-axis.
Step by step solution
01
Identify the Type of Conic Section
Recognize that the given equation \( y^2 - \frac{x^2}{k^2} = 1 \) is an equation of a hyperbola. This is because it has a subtraction of terms with squares, and the constant on the right side is positive.
02
Understand the Hyperbola's Axes Orientation
Note that the equation \( y^2 - \frac{x^2}{k^2} = 1 \) represents a hyperbola that opens upwards and downwards because the \( y^2 \) term is positive, indicating a vertical transverse axis.
03
Recognize the Role of Parameter \(k\)
The parameter \( k \) appears in the denominator of the \( x^2 \) term in the equation \( \frac{x^2}{k^2} \). The magnitude of \( k \) affects the spread of the hyperbola along the x-axis.
04
Analyze the Effect of Increasing \(|k|\)
Increasing \(|k|\) means that \( k^2 \) increases, making \( \frac{x^2}{k^2} \) smaller for any given \( x \). As \(|k|\) becomes larger, the hyperbola becomes narrower along the x-axis, as the x-values contributing to the hyperbola's shape become closer to zero.
05
Summarize the Change in Graph
As \(|k|\) increases, the graph of the hyperbola \( y^2 - \frac{x^2}{k^2} = 1 \) becomes narrower, or more 'vertical', across its branches because the spread along the x-axis decreases. This 'squeezing' effect implies more emphasis on the term unaffected by \( k \), which is \( y^2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbola
A hyperbola is a distinct type of conic section, which looks like two separate, mirror-image curves, called branches, that open in opposite directions. Unlike circles or ellipses, a hyperbola is defined by the subtraction in its standard equation form: \( y^2 - \frac{x^2}{k^2} = 1 \). This subtraction is what sets apart hyperbolas from other conic sections. In this equation, you have one positive and one negative squared term, indicating the shape's unique structure.
- The branches of a hyperbola can either open horizontally or vertically, depending on the placement of the positive and negative terms.
- Hyperbolas are often portrayed graphically with their asymptotes, which are straight lines that the branches approach but never meet.
- The constant on the right side of the equation equal to 1 signifies the standard form, which is useful for immediately recognizing its type.
Parameter Effects
Parameters in a hyperbola equation like \( y^2 - \frac{x^2}{k^2} = 1 \) can significantly influence its appearance. Here, the parameter \( k \) directly affects the width of the hyperbola's branches. When you change \( k \), you are modifying how far or close the branches are spread apart from the center on the x-axis.
- When \( |k| \) increases, it means \( k^2 \) becomes larger. This causes the component \( \frac{x^2}{k^2} \) to decrease for the same x-values, essentially squeezing the hyperbola narrower.
- The hyperbola's overall shape will appear more 'compressed' along the x-axis, concentrating more around the y-axis.
- This effect illustrates how parameter changes can alter the geometry of conic sections, allowing for a wide range of graphical representations.
Axes Orientation
Axes orientation in hyperbolas determines the direction in which the branches open. In the equation \( y^2 - \frac{x^2}{k^2} = 1 \), because \( y^2 \) is the positive term, the transverse axis鈥攚here the hyperbola's branches open鈥攊s vertical. This means the branches stretch upwards and downwards from the center.
- If the equation were \( x^2 - \frac{y^2}{k^2} = 1 \), the transverse axis would be horizontal, resulting in branches opening to the left and right.
- The knowledge of axes orientation is crucial for sketching the graph, as it immediately suggests the spatial alignment of the hyperbola.
- In vertical hyperbolas, the vertices (the points closest to the center) directly indicate this vertical stretch.
