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Chapter 10: Problem 3

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$\frac{y^{2}}{9}-\frac{x^{2}}{4}=1$$

Short Answer

Expert verified
Vertices: (0, 3) and (0, -3); Foci: (0, 鈭�13) and (0, -鈭�13); Asymptotes: y = (3/2)x and y = -(3/2)x.

Step by step solution

01

Recognize the Hyperbola Equation Form

The given equation is \(\frac{y^2}{9} - \frac{x^2}{4} = 1\). This is in the standard form for a vertical hyperbola, \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Here, \(a^2 = 9\) and \(b^2 = 4\).
02

Identify the Vertices

For a vertical hyperbola with equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the vertices are at \((0, \pm a)\). Since \(a^2 = 9\), we have \(a = 3\). Thus, the vertices are at \((0, 3)\) and \((0, -3)\).
03

Calculate the Foci

The foci for a vertical hyperbola are found using the formula \((0, \pm c)\), where \(c^2 = a^2 + b^2\). Here, \(c^2 = 9 + 4 = 13\), so \(c = \sqrt{13}\). Thus, the foci are located at \((0, \sqrt{13})\) and \((0, -\sqrt{13})\).
04

Derive the Equations of Asymptotes

The asymptotes for a vertical hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) have equations \(y = \pm \frac{a}{b}x\). Substituting \(a = 3\) and \(b = 2\) gives \(y = \pm \frac{3}{2}x\).
05

Sketch the Hyperbola

To sketch the hyperbola, plot the vertices \((0, 3)\) and \((0, -3)\), the foci \((0, \sqrt{13})\) and \((0, -\sqrt{13})\). Draw the asymptotes \(y = (3/2)x\) and \(y = -(3/2)x\) as dashed lines. The hyperbola branches open upwards and downwards along the y-axis and approach the asymptotes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertices of a Hyperbola
In a hyperbola, the vertices are the points where the hyperbola intersects its axis. For a vertical hyperbola, these points lie along the y-axis. If you have the standard hyperbola equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), you can easily identify the vertices using \( a \), which is derived from \( a^2 \). In our example:
  • Given \( a^2 = 9 \), we find \( a = 3 \).
  • The vertices are at \( (0, a) \) and \( (0, -a) \), thus at \( (0, 3) \) and \( (0, -3) \).
These points are essential as they help define the shape and size of the hyperbola. Knowing where the vertices lie gives us a starting point for sketching the graph.
Foci of a Hyperbola
The foci (plural for focus) of a hyperbola are critical points used to define and shape the graph. For a vertical hyperbola, the foci lie along the same line as the vertices, which is the y-axis in our case. The formula to locate the foci is \( (0, \pm c) \) for a vertical hyperbola, with \( c \) calculated using:
  • \( c^2 = a^2 + b^2 \)
For our specific hyperbola,
  • \( a^2 = 9 \) and \( b^2 = 4 \), so \( c^2 = 13 \).
  • This means \( c = \sqrt{13} \).
  • The foci are positioned at \( (0, \sqrt{13}) \) and \( (0, -\sqrt{13}) \).
These points help further define the width of the branches of the hyperbola as they spread away from the vertices.
Equations of Asymptotes
Asymptotes in a hyperbola are the straight lines that the curves approach but never reach. They are crucial guides for plotting the hyperbola accurately. For a vertical hyperbola in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the asymptotes have the equations
  • \( y = \pm \frac{a}{b} x \).
For our example:
  • Given \( a = 3 \) and \( b = 2 \), substitute into the formula: \( y = \pm \frac{3}{2} x \).
These asymptotes are imaginary bounds that help us sketch the hyperbola, showing the direction towards which each branch opens and extends.
Graphing Hyperbolas
Graphing a hyperbola involves understanding its vertices, foci, and asymptotes, as they guide the entire plotting process. By placing these elements accurately on a graph:
  • Start by plotting the vertices at \( (0, 3) \) and \( (0, -3) \).
  • Next, mark the foci located at \( (0, \sqrt{13}) \) and \( (0, -\sqrt{13}) \).
  • Draw dashed lines for the asymptotes, \( y = \frac{3}{2}x \) and \( y = -\frac{3}{2}x \).
With these elements plotted, you can sketch the hyperbola, displaying the branches of the curve opening upwards and downwards along the y-axis, naturally turning towards the asymptotes as they extend. This approach captures the essence of the hyperbola's shape and behavior.

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