Problem 63 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated part of the hyperbola. Upper halves of the branches of \(\frac{x^{2}}{9}-\frac{y^{2}}{36}=1\)

Step-by-Step Solution

Answer

The equation for the upper halves is \( y = 6 \sqrt{\frac{x^2}{9} - 1} \).

Step 1: Understand the Hyperbola Equation

The given equation \( \frac{x^2}{9} - \frac{y^2}{36} = 1 \) is the equation of a hyperbola. This equation describes a hyperbola centered at the origin with the transverse axis along the x-axis.

Step 2: Rearrange to Solve for y^2

To find the equation for the upper halves of the branches, rearrange the hyperbola equation to solve for \( y^2 \). Start by isolating \( y^2 \): \[ \frac{y^2}{36} = \frac{x^2}{9} - 1 \] Then multiply both sides by 36 to solve for \( y^2 \): \[ y^2 = 36 \left( \frac{x^2}{9} - 1 \right) \]

Step 3: Find the Expression for y

Take the square root of both sides to solve for \( y \), noting that since we want the upper half, we will use the positive square root: \[ y = \sqrt{36 \left( \frac{x^2}{9} - 1 \right)} \] Simplifying further, this becomes: \[ y = 6 \sqrt{\frac{x^2}{9} - 1} \]

Step 4: Simplify the Equation for the Upper Halves

Since \( y = 6 \sqrt{\frac{x^2}{9} - 1} \) represents the upper half of the hyperbola branches, the final equation for the upper halves is simplified as \( y = 6 \sqrt{\frac{x^2}{9} - 1} \).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four primary types are circles, ellipses, parabolas, and hyperbolas. Each type of conic is defined by its unique equation. These sections are crucial in various mathematical and scientific applications, ranging from astronomy to engineering.

Circles and ellipses are formed from slices perpendicular or at an angle to the cone's axis that do not pass through its vertex. Parabolas result from a parallel slice to one of the cone's nappes, while hyperbolas are formed when the plane intersects both nappes.

• Circle: All points are equidistant from a central point.
• Ellipse: A stretched circle with two focal points.
• Parabola: Open curve with a single focus and directrix.
• Hyperbola: Consists of two branches, each approaching asymptotically.

Understanding the nature of conic sections, like the hyperbola in this exercise, provides insight into the geometric properties of these fascinating shapes.

Transverse Axis

In the context of hyperbolas, the transverse axis is essentially the 'spine' or main axis of the hyperbola. It's the line that passes through both foci (the central points of focus for each branch) and the center of the hyperbola.

For hyperbolas represented by equations like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis runs horizontally along the x-axis if \(a^2\) is under the \(x^2\) term, as in our problem. This signifies that it stretches across the x-direction.

• The transverse axis determines the distance between the vertices of the hyperbola.
• It's pivotal because it aligns with the direction of the hyperbola's branches.
• The length of our step-by-step solution's transverse axis is \(2a\), or in this problem, \(2 \times 3 = 6\).

Recognizing the transverse axis helps when graphing the hyperbola and analyzing its geometric properties.

Hyperbola Branches

A hyperbola consists of two distinct and separate curves, known as branches. Each branch is a mirror image of the other with respect to the center of the hyperbola.

These branches are characterized by their unique, open-ended shape, and they stretch continuously away from each other. The open parts of the branches point towards the asymptotes of the hyperbola, which are imaginary lines the branches approach but never touch.

• Each branch is equidistant from the center but open in opposite directions.
• The branches get infinitely closer to their asymptotes, therefore appearing "open."
• The direction of opening depends on the transverse axis; in this case, they open horizontally.

In this exercise, understanding hyperbola branches helps you identify which parts of the hyperbola relate to the problem's specified regions—you work with the upper halves of these branches specifically.

Upper Half of Hyperbola

When solving the equation for the upper halves of a hyperbola, you're focusing on one specific part of each branch: the part that extends above the transverse axis.

In mathematical terms, the upper half of each branch is found by taking the positive square root when solving for \(y\). This is represented by equations like \(y = 6 \sqrt{\frac{x^2}{9} - 1}\), derived from our original hyperbola equation.

• Each branch has an upper and lower half, but solutions can target just the upper half, as in this case.
• The focus is only on the positive \(y\) values.
• This makes the graphing of equations much more manageable when seeking only the topmost regions.

Recognizing the concept of the upper half is crucial for completing complex geometry assignments, as it narrows down from a full plot to just part of the graphical representation.