Problem 15 - Algebra 2

Question

Graph each equation. Identify the conic section and describe the graph and its lines of symmetry. Then find the domain and range. $$ 4 y^{2}-36 x^{2}=1 $$

Step-by-Step Solution

Answer

The given equation represents a vertical hyperbola with lines of symmetry along the vertical axis (x = 0). The domain and range are both all real numbers, so the domain is (-∞, ∞) and the range is (-∞, ∞).

Step 1: Identify the Conic Section

The given equation has both x and y variables squared but with different coefficients and signs. Thus, it describes a hyperbola. Since the coefficient of \(y^2\) is positive and the coefficient of \(x^2\) is negative, it’s a vertical hyperbola.

Step 2: Rewrite in Standard Form

Rewrite the given equation in the standard form for vertical hyperbolas by factoring out the coefficients from the square terms and setting it equal to 1: \[\frac{y^2}{(\frac{1}{4})} - \frac{x^2}{(\frac{1}{36})} = 1\] or \[\frac{y^2}{\frac{1}{4}} - \frac{x^2}{\frac{1}{36}} = 1\]

Step 3: Describe the Graph and Identify the Lines of Symmetry

The hyperbola opens upwards and downwards (since \(y^2\) is positive). It has a vertical axis of symmetry (x = 0). There are two branches, occurring above and below the x-axis, reflecting across the line x = 0.

Step 4: Find the Domain and Range

For a hyperbola, the domain (set of all possible x-values) and the range (all possible y-values) are all real numbers. Therefore, the domain is (-∞, ∞) and the range is (-∞, ∞).

Key Concepts

Hyperbola

A hyperbola is a fascinating conic section formed by intersecting a double cone with a plane. Unlike circles and ellipses, which enclose areas, a hyperbola consists of two separate, symmetrical curves called branches. These branches open outward, portrayed as mirror images across the horizontal or vertical axis. In mathematics, a hyperbola is typically defined by equations of the form:

• Horizontal hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertical hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

In our exercise, the equation \(4y^2 - 36x^2 = 1\) was identified as a vertical hyperbola. This conclusion is based on understanding the signs of the squared terms: the \(y^2\) term is positive, indicating a vertical opening. In simpler terms, this hyperbola stretches in the up and down direction rather than side to side, forming an hourglass shape.
Understanding hyperbolas is essential in various fields, such as astronomy, where the paths of celestial bodies sometimes follow hyperbolic trajectories.

Domain and Range

In any mathematical expression, the domain and range provide critical insights into the function's behavior and possibilities.
The domain of a function consists of all possible x-values that will yield real y-values. For a hyperbola, the domain is usually all real numbers unless restricted by added conditions, like the specific bounds of a region of interest.
In our example, the hyperbola \(4y^2 - 36x^2 = 1\) has a domain of \((-\infty, \infty)\). This means the x-value can be any real number, highlighting the positions along the hyperbola's transverse axis
The range of a function, on the other hand, refers to all possible y-values that correspond to the x-values in the domain. Similarly, for the given hyperbola, the range is \((-\infty, \infty)\). This suggests that the hyperbola extends indefinitely in both vertical directions.
Understanding domain and range helps us comprehend the expansiveness or limitations of a hyperbola's graph.

Lines of Symmetry

Lines of symmetry are axes along which a shape can be folded to create two identical halves. They are crucial in understanding the symmetrical properties and orientation of a conic section.
In the case of hyperbolas, identifying lines of symmetry helps reveal the principal directions in which the curves extend. Generally, a hyperbola displays two lines of symmetry:

• One following the transverse axis (where the branches open).
• Another that is perpendicular to the transverse axis, known as the conjugate axis.

For the given hyperbola \(4y^2 - 36x^2 = 1\), the line of symmetry aligns vertically along \(x = 0\). This vertical axis is the line over which the two hyperbola branches are reflected, ensuring each half is a mirror image of the other.
Such symmetrical insights are beneficial not only in plotted graphs but also in physical phenomena, such as when designing lenses and reflectors that utilize hyperbolic shapes for precise light and wave manipulation.