Problem 32 - Thomas Calculus

Question

Graph the upper branch of the hyperbola \(y^{2}-16 x^{2}=1\)

Step-by-Step Solution

Answer

Sketch the hyperbola's upper branch using its vertex at (0, 1) and asymptotes y = ±(1/4)x.

Step 1: Identify the Standard Form

The given equation is \(y^2 - 16x^2 = 1\). The standard form of a hyperbola centered at the origin with a vertical transverse axis is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Here, \(a^2 = 1\) and \(b^2 = 16\), so the standard form is correct.

Step 2: Calculate Vertices

For the hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the vertices are at \((0, \pm a)\). Here, since \(a^2 = 1\), \(a = 1\). Thus, the vertices are \((0, 1)\) and \((0, -1)\). For the upper branch, we consider the vertex \((0, 1)\).

Step 3: Determine Asymptotes

The asymptotes for the given hyperbola are calculated using the equation \(y = \pm\frac{a}{b}x\). Substituting \(a = 1\) and \(b = 4\) (since \(b^2 = 16\)), the equations of the asymptotes are \(y = \pm\frac{1}{4}x\).

Step 4: Sketch the Graph

On a graph with axes, plot the vertex \((0, 1)\) and sketch the asymptotes \(y = \frac{1}{4}x\) and \(y = -\frac{1}{4}x\). Draw the upper branch of the hyperbola, starting from the vertex \((0, 1)\) and curving towards the asymptotes without touching them.

Key Concepts

Standard Form of Hyperbola

A hyperbola is a type of conic section that can be described by its standard form equation. Understanding this form is crucial for sketching and analyzing hyperbolas. The standard form of a hyperbola that opens vertically, and is centered at the origin, is expressed as:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]In this setup, "a" and "b" are real numbers that help define the geometry of the hyperbola. Specifically, "a" represents the semi-major distance along the y-axis, while "b" signifies the semi-distance that helps define the hyperbola's shape. For the given equation, \(y^2 - 16x^2 = 1\), we can see that it matches the standard form with \(a^2 = 1\) and \(b^2 = 16\). Simplifying gives us \(a = 1\) and \(b = 4\). This understanding is the foundation for finding other key features like vertices and asymptotes.

Vertices of Hyperbola

Vertices are critical points that define a hyperbola's shape. They denote the closest points of the hyperbola to its center. In the standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the vertices for a vertically oriented hyperbola are located at \((0, \pm a)\).
For our equation, \(a = 1\), so the vertices are positioned at \((0, 1)\) and \((0, -1)\). These points are plotted along the y-axis because of the vertical orientation of the hyperbola. For graphing the upper branch of the hyperbola, we use the vertex \((0, 1)\). It helps in anchoring the curve of the hyperbola and guiding the sketch towards the asymptotes.

Asymptotes of Hyperbola

Asymptotes are imaginary lines that the branches of a hyperbola approach but never intersect. They create a framework that the curve of the hyperbola clings to without touching. For a hyperbola given by the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the asymptotic equations are given by:\[ y = \pm \frac{a}{b}x \]In our case, where \(a = 1\) and \(b = 4\), the asymptotes are \(y = \pm \frac{1}{4}x\). These lines are drawn through the origin (0,0) and have slopes of \(\frac{1}{4}\) and \(-\frac{1}{4}\). Drawing these lines can help guide the sketch of the hyperbola branches, forming a cross-like template that the branches hug closely without intersecting. They are essential for drawing an accurate hyperbola.

Hyperbola Graph

Graphing a hyperbola involves sketching the vertices, asymptotes, and then the branches themselves. To begin, place the vertices \((0, 1)\) and \((0, -1)\) onto a Cartesian plane. Use these points as anchors. Next, draw the asymptotes \(y = \frac{1}{4}x\) and \(y = -\frac{1}{4}x\). These should extend diagonally through the origin, creating an 'X' shape.Once the vertices and asymptotes are in place, proceed to sketch the upper branch of the hyperbola. Start at the vertex \((0, 1)\) and smoothly curve out towards the asymptotes without crossing them. The curve should appear to "escape" towards the asymptotes while staying between them, showcasing its characteristic long, narrow U-shape for the upper branch. By following these steps, you can accurately represent the hyperbola graphically.