Problem 30 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the hyperbola. $$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$

Step-by-Step Solution

Answer

Graph the equation with vertices at (10,0) and (-10,0), foci at (±12.81,0), and asymptotes y=±(4/5)x.

Step 1: Identify Hyperbola Standard Form

The given equation is \( \frac{x^2}{100} - \frac{y^2}{64} = 1 \). This is in the standard form of a horizontal hyperbola, \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \( a^2 = 100 \) and \( b^2 = 64 \).

Step 2: Calculate the Semi-Axes Lengths

From \( a^2 = 100 \), we find \( a = \sqrt{100} = 10 \). From \( b^2 = 64 \), we find \( b = \sqrt{64} = 8 \).

Step 3: Determine the Vertices

For a horizontal hyperbola, the vertices are at \( (\pm a, 0) \). Hence, the vertices are \( (10, 0) \) and \( (-10, 0) \).

Step 4: Locate the Foci

The foci of a hyperbola are given by \( c = \sqrt{a^2 + b^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81 \). Thus, the foci are located at approximately \( (\pm 12.81, 0) \).

Step 5: Sketch the Asymptotes

The equations of the asymptotes for a horizontal hyperbola are \( y = \pm \frac{b}{a}x \). Here, \( y = \pm \frac{8}{10}x \) or equivalently \( y = \pm \frac{4}{5}x \).

Step 6: Draw the Hyperbola

Using a graphing device, input the hyperbola equation \( \frac{x^2}{100} - \frac{y^2}{64} = 1 \) and plot the vertices, foci, and asymptotes. Ensure the hyperbola opens left and right along the x-axis through the vertices, approaching but never crossing the asymptotes.

Key Concepts

Conic Sections

A conic section is a curve obtained by intersecting a plane with a double-napped cone. There are four main types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type is defined by the angle of intersection with the cone and varies distinctively by its equation.
Hyperbolas, one type of conic section, are particularly interesting as they have two distinct, separate curves. These curves are mirror images of each other. This is unlike circles and ellipses, which are closed curves, and unlike parabolas, which open in a single direction. Hyperbolas open outwards in opposite directions and can be oriented either horizontally or vertically based on how they intersect the cone.
Recognizing and understanding the nature of hyperbolas as conic sections is fundamental in studying these fascinating geometric shapes.

Standard Form of Hyperbola

The standard form of a hyperbola’s equation helps us easily identify its orientation and characteristics. There are two forms based on whether the hyperbola is horizontal or vertical. For a horizontal hyperbola, the equation looks like this:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Meanwhile, a vertical hyperbola uses this form: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

Here, \( a^2 \) and \( b^2 \) represent positive real numbers that define the hyperbola’s radii along its axes. Specifically, \( a \) relates to the distance from the center to the vertices, and \( b \) helps determine the distance associated with the asymptotes.
These equations allow us to find critical elements of the hyperbola like its vertices, foci, and asymptotes, which define its shape and position in the coordinate plane.

Semi-Axes Lengths

Semi-axes lengths are crucial parameters in understanding a hyperbola. These lengths consist of the transverse axis and the conjugate axis, represented by \( a \) and \( b \) respectively.
The transverse axis of a hyperbola is equivalent to its principle length. It directly runs through both foci, and its length is denoted by \( 2a \). Calculating \( a \) comes from the equation's denominator associated with \( x^2 \) or \( y^2 \), depending on the orientation.

• For example, in the equation \( \frac{x^2}{100} - \frac{y^2}{64} = 1 \), we find \( a = \sqrt{100} = 10 \).

Similarly, the conjugate axis length is found by \( 2b \), derived from the equation's remaining denominator. Here, \( b = \sqrt{64} = 8 \). The semi-axes work together to dictate the size and spread of the hyperbola’s open branches.

Vertices and Foci of Hyperbola

Understanding the vertices and foci of a hyperbola provides valuable insight into its graph. The vertices are the points where the hyperbola intersects the transverse axis. For a horizontal hyperbola, these are located at \( (\pm a, 0) \), while in a vertical setup, it’s \( (0, \pm a) \).
In the example equation \( \frac{x^2}{100} - \frac{y^2}{64} = 1 \), the vertices are at \( (10, 0) \) and \( (-10, 0) \).

• Next, the foci provide additional insight into the curvature. For a hyperbola, the foci amplifies its openness. The distance to each focus from the center is given by \( c = \sqrt{a^2 + b^2} \). With \( c \approx 12.81 \), the foci for this hyperbola are approximately \( (\pm 12.81, 0) \).

The positions of vertices and foci help describe the extent and direction in which the hyperbola opens, important for sketching and understanding its nature.