Problem 6 - Algebra and Trigonometry

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\)

Step-by-Step Solution

Answer

Vertices: (0, 3) and (0, -3); Foci: (0, 5) and (0, -5); Asymptotes: y = \(\pm\frac{3}{4}x\).

Step 1: Identify the General Form

The given equation \(\frac{y^2}{9} - \frac{x^2}{16} = 1\) is in the standard form of a vertical hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where \(a^2 = 9\) and \(b^2 = 16\). Hence, \(a = 3\) and \(b = 4\).

Step 2: Find the Vertices

For a vertical hyperbola, the vertices are at \((0, \pm a)\). Since \(a = 3\), the vertices are \((0, 3)\) and \((0, -3)\).

Step 3: Calculate the Foci

The foci of a vertical hyperbola are found at \((0, \pm c)\), where \(c^2 = a^2 + b^2\). Here, \(c^2 = 9 + 16 = 25\); thus, \(c = 5\). Therefore, the foci are \((0, 5)\) and \((0, -5)\).

Step 4: Determine the Asymptotes

The asymptotes of a vertical hyperbola are given by \(y = \pm \frac{a}{b} x\). In this case, \(\frac{a}{b} = \frac{3}{4}\). Thus, the equations of the asymptotes are \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\).

Step 5: Sketch the Graph

To sketch the graph, plot the vertices at \((0, 3)\) and \((0, -3)\). Draw the asymptotic lines \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\) as dashed lines through the origin. Plot the foci inside the hyperbola at \((0, 5)\) and \((0, -5)\). Sketch the hyperbola curves opening upwards and downwards, approaching the asymptotes but never touching them.

Key Concepts

Vertices of Hyperbola

Vertices are crucial points on a hyperbola, forming the main boundary from which the curves extend. For a vertical hyperbola (when the squared term with the positive coefficient is in the numerator for the vertical line of symmetry), such as given \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \), the vertices are positioned along the y-axis.

• Their general form is \((0, \pm a)\) where \(a\) is derived from \(a^2\). In our equation, \(a^2 = 9\), therefore \(a = 3\).
• This means the vertices are at \((0, 3)\) and \((0, -3)\).

By identifying the vertices, we understand the basic extent of the hyperbola along the axis perpendicular to the transverse axis.

Foci of Hyperbola

The foci of a hyperbola are vital points that lie along the axis of symmetry and are tucked inside the curves of the hyperbola. They help define the hyperbola's shape and eccentricity. To find these foci, we use the relationship involving \(c\), where \(c^2 = a^2 + b^2\).

• Here, \(a^2 = 9\) and \(b^2 = 16\), resulting in \(c^2 = 25\) and \(c = 5\).
• The foci are thus located at \((0, \pm c) = (0, 5)\) and \((0, -5)\).

The foci are always placed closer together as compared to the vertices, and these are points where the distances were used when formulating the hyperbola equation.

Asymptotes of Hyperbola

Asymptotes are straight lines that the hyperbola approaches but never meets or intersects with. These lines guide the curvature of the hyperbola and give us a sense of its openness. For the standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the asymptotes are given by the equation:

• \(y = \pm \frac{a}{b} x\).
• Here, \(\frac{a}{b} = \frac{3}{4}\), hence the equations of asymptotes are \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\).

These lines run diagonally through the origin and help in sketching the hyperbola accurately. Plotting these asymptotes can dramatically enhance the accuracy of your hyperbola's graph.

Vertical Hyperbola

A vertical hyperbola is identified by its vertical transverse axis. This means that the hyperbola opens upwards and downwards, unlike a horizontal hyperbola which stretches side to side. This distinct direction is recognized by examining the placement of the terms in the equation:

• In a vertical hyperbola, the "\(y\)" term leads in the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
• This configuration tells us that the vertices and foci will lie vertically along the y-axis.

Navigating vertical hyperbolas involves focusing on the y-orientation for symmetry, vertices, and foci, making it one of the main characteristics that affect the hyperbola's appearance on a graph.

Graphing Hyperbolas

Graphing a hyperbola primarily involves locating and marking its key features like vertices, foci, and asymptotes.Follow these steps to efficiently graph your hyperbola:

• Start by plotting the vertices on the graph. For our vertical hyperbola, use \((0, 3)\) and \((0, -3)\).
• Show the asymptotes as dashed lines through the origin with original equations like \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\).
• Position the foci within the curves at \((0, 5)\) and \((0, -5)\).
• Draw the hyperbola curves so they open in the vertical direction, approaching but never touching the asymptotes.

Ensuring this step-by-step graphing will help provide a clearer picture of the hyperbola and its subtleties, capturing its full structure and form.