Problem 39 - Algebra 2

Question

Describe how the graph of \(y^{2}-\frac{x^{2}}{k^{2}}=1\) changes as \(|k|\) increases.

Step-by-Step Solution

Answer

As \(|k|\) increases, the hyperbola becomes narrower along the x-axis.

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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 \( y^2 \).

Key Concepts

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.

Understanding this basic structure is useful in visualizing what changes will occur when you adjust different parameters in its equation.

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.

By manipulating \( k \), you can explore how the width of a hyperbola responds, helping you gain a deeper understanding of its graph.

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—where the hyperbola's branches open—is 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.

Recognizing the orientation not only aids in graphing but also helps in interpreting the effects of parameter changes like \( k \). This understanding provides a clearer picture of how conic sections transform on a plane.