Problem 66 - Precalculus: Functions and Graphs

Question

Describe the part of a hyperbola given by the equation. $$x=-\frac{5}{4} \sqrt{y^{2}+16}$$

Step-by-Step Solution

Answer

The hyperbola opens horizontally; its part is defined for negative x-values.

Step 1: Identify the Hyperbola Equation

The given equation is \( x = -\frac{5}{4} \sqrt{y^2 + 16} \). This is a form of a hyperbola equation, where \( x \) is expressed in terms of \( y \).

Step 2: Rewrite the Equation in Standard Form

The standard form of a hyperbola can be expressed as \( x^2/a^2 - y^2/b^2 = 1 \) or the reverse. In this problem, it's expressed differently. Let's transform it:\[ x = -\frac{5}{4} \sqrt{y^2 + 16} \] Try to isolate \( \sqrt{y^2 + 16} \):\[ \frac{-4x}{5} = \sqrt{y^2 + 16} \]

Step 3: Square Both Sides

Removing the square root by squaring both sides yields:\[ \left(\frac{-4x}{5}\right)^2 = y^2 + 16 \]Simplifying gives:\[ \frac{16x^2}{25} = y^2 + 16 \]

Step 4: Simplify to Find the Relation

Now, simplify and solve for \( y^2 \):\[ \frac{16x^2}{25} - 16 = y^2 \] Express in terms of a common denominator:\[ \frac{16x^2 - 400}{25} = y^2 \]

Step 5: Determine the Orientation and Features

This is a rearrangement involving \( x^2 \) and \( y^2 \), indicative of the structure of a hyperbola. As the y-components are positive when re-organized, it shows the hyperbola opens horizontally, and compared against the standard hyperbola orientation, the absence of cross terms implies no rotation.

Key Concepts

Graphing Hyperbolas

Graphing hyperbolas may seem complex at first, but breaking it down into steps can make it easier to understand. A hyperbola is a type of conic section, defined as the set of all points where the difference of the distances to two fixed points (the foci) is constant. When graphing, ensuring the equation is in a recognizable format is the first key step.
To successfully graph a hyperbola, it is essential to identify its center, vertices, and asymptotes:

• Center: The midpoint at which the hyperbola is centered. Often seen in the equations as (h, k).
• Vertices: Located along the axis of symmetry passing through the center.
• Asymptotes: Diagonal lines the hyperbola approaches but never intersects. These are derived from the slopes found in the equation.

The given equation in the form of \( x = -\frac{5}{4} \sqrt{y^2 + 16} \) is slightly different from the conventional form. In this instance, we transform and rearrange to understand its components and features, ensuring it follows the general pattern of a hyperbola graph. Familiarity with both the horizontal and vertical orientation equations will assist in accurately plotting the hyperbola, with this particular form suggesting a horizontal orientation.

Transformation of Equations

When tackling hyperbola equations, being able to manipulate and transform the expression is crucial. The standard form for hyperbolas is typically \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). Understanding how to restructure the equation into these forms reveals much more about its characteristics.
The transformation process involves several algebraic steps, often as follows:

• Isolate components: Rearrange the given equation to isolate the hyperbolic components \( x \) and \( y \) in terms of standard variables.
• Squaring and simplification: Square both sides to eliminate square roots, thereby simplifying the equation.
• Common denominators: Dealing with fractions by determining a common denominator helps in reformatting the terms for clarity.

In our problem, the transformation involved changing the given equation \( x = -\frac{5}{4} \sqrt{y^2 + 16} \) into a more recognizable form to derive the relationship between \( x^2 \) and \( y^2 \). This step is crucial in identifying the features and orientation of the hyperbola.

Conic Sections

Hyperbolas are one of the four types of conic sections, along with circles, ellipses, and parabolas. Conic sections are curves obtained by intersecting a cone with a plane. Each type of conic section has unique properties and equations.
Here's a quick breakdown:

• Circle: A set of points at a constant distance from a center point.
• Ellipse: A set of points where the sum of the distances from two foci is constant.
• Parabola: A set of points where each point is equidistant from a single focus and a directrix.
• Hyperbola: Involves a difference in distance from each point to two foci remaining constant.

Hyperbolas are unique in terms of having two disconnected curves, or branches, with properties and appearances quite distinct from the other conic forms. Their defining eccentricity is greater than one, highlighting their wider spread compared to ellipses. Understanding how hyperbolas operate within the broader category of conics provides a deeper comprehension of their mathematical behavior and graphical representations.