Problem 9 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Recognize the Form of the Equation

The equation \(\frac{x^2}{4} - \frac{y^2}{16} = 1\) is in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This form is known from conic sections. It's important to identify which conic section this form represents.

Step 2: Identify the Type of Conic Section

The equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) corresponds to the standard form of a horizontal hyperbola. A hyperbola has a subtraction sign between the terms, distinguishing it from ellipses which have a plus sign.

Step 3: Conclusion about the Conic Section

Based on the recognition in Step 1 and the identification in Step 2, we conclude that the equation represents a hyperbola. No graph is required for this conclusion.

Key Concepts

Hyperbola

A hyperbola is one of the four main types of conic sections—the others being the circle, ellipse, and parabola. What sets hyperbolas apart is their distinct geometric shape, defined by two open curves. These curves are mirror images of each other.
Hyperbolas appear when you cut a double cone (two cones stacked point to point) with a plane. Unlike ellipses, which resemble closed loops, hyperbolas are open-ended shapes and go off to infinity.

• Hyperbolas have two branches, which bow away from each other.
• The overlapping plane does not pass through the cone at a point or in parallel to its base, unlike the other conic sections.
• The main feature of a hyperbola is its vertices and foci. The vertices are the closest points of each arm of the hyperbola to each other.
• Each arm flares out around a fixed point known as the focus of the hyperbola.

Understanding hyperbolas becomes easier when you also grasp the idea of the center and the axes of symmetry. For hyperbolas, lines called asymptotes run through the center and approach closely but never intersect a hyperbolic curve. These lines give form to the shape of the hyperbola.

Standard Form

The standard form of a hyperbola is a key aspect of recognizing these shapes without graphing. Hyperbolas can be horizontal or vertical, and their equations reflect this.
For a horizontal hyperbola, the standard form appears as:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
While for a vertical hyperbola, it's represented by:
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]

• In these equations, "a" and "b" are constants that determine the shape and orientation of the hyperbola.
• The key to finding the orientation is looking for whether the "x" term or "y" term comes first.
  If the "x" term comes first, the hyperbola is horizontal.
• If the "y" term comes first, it's vertical.

Moreover, seeing the minus sign between the two fractions is a quick indicator that we are dealing with a hyperbola. This is what differentiates it from an ellipse which uses a plus sign.

Equation Recognition

Equation recognition is a valuable skill that enables you to identify the type of conic section represented by an equation without the need for graphing. Being able to recognize the form \[\frac{x^2}{4} - \frac{y^2}{16} = 1\]immediately points to it being a hyperbola based on the standard forms we discussed earlier.

• First, take note of the negative sign separating the two fractions; this indicates a hyperbola.
• Next, compare the equation to the standard form for hyperbolas. We notice the structure \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\].
• This matches our equation, revealing it's a horizontal hyperbola because the "x" term appears first.

This ability to recognize equations expands your analytical skills in mathematics. It helps you deduce properties of the hyperbola such as orientation, center, and axes without plotting points on a graph.