Problem 26 - College Algebra

Question

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{y^{2}}{3^{2}}-\frac{x^{2}}{3^{2}}=1 $$

Step-by-Step Solution

Answer

The equations of the asymptotes are \( y = x \) and \( y = -x \).

Step 1: Identify the standard form of the hyperbola equation

The given equation is \( \frac{y^2}{3^2} - \frac{x^2}{3^2} = 1 \). This is in the standard form for a vertical hyperbola: \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), where \( a = 3 \) and \( b = 3 \).

Step 2: Recall the formula for the asymptotes of a vertical hyperbola

For a vertical hyperbola \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), the asymptotes have the equations \( y = \pm \frac{b}{a} x \).

Step 3: Substitute the values of a and b into the asymptotes' formula

Substituting \( a = 3 \) and \( b = 3 \) into the formula, we get \( y = \pm \frac{3}{3} x \), which simplifies to \( y = \pm x \).

Key Concepts

Standard Form of a Hyperbola

A hyperbola is a type of conic section that can be represented in a standardized way referred to as its standard form. Understanding the standard form is crucial for determining the properties and characteristics of the hyperbola, including its asymptotes. The standard form of a hyperbola depends on whether it opens vertically or horizontally.

The given exercise features a hyperbola in standard form for vertical orientation:

• For a vertical hyperbola, the standard form is:\[\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\]
• This format highlights that the hyperbola opens upwards and downwards from the center.
• The values of \(a\) and \(b\) determine the size and shape of the hyperbola. It's important to distinguish \(a^2\) and \(b^2\) by their respective positions: \(y^2\) over \(b^2\) and \(x^2\) over \(a^2\).

Identifying and understanding the standard form sets the stage for finding more information such as the asymptotes. Each component in the standard form contributes to the hyperbola's geometry.

Vertical Hyperbola

In a vertical hyperbola, the main axis along which the hyperbola extends is vertical. This means the hyperbola opens up and down rather than left and right. The standard form for a vertical hyperbola is \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).

• In the equation \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \), both \(a^2\) and \(b^2\) are equal, indicating that the hyperbola components are symmetric around both axes.
• This symmetry can provide a simpler equation for asymptotes, as seen with\(a = b\).

Another key feature of vertical hyperbolas is their asymptotes, which we will delve into next. Knowing whether a hyperbola is vertical or horizontal helps you apply the correct formula to find the equations of the asymptotes.

Equation of Asymptotes

Asymptotes are lines that a hyperbola approaches but never actually touches. They define the shape of the hyperbola's open ends and can be found using the hyperbola's equation. For a vertical hyperbola, the asymptotes have the formula \( y = \pm \frac{b}{a} x \).

• Substitute \(a\) and \(b\) from the standard equation \( \frac{y^2}{3^2} - \frac{x^2}{3^2} = 1 \). Here, \(a = 3\) and \(b = 3\), so the asymptotes' equation simplifies to \( y = \pm x \).
• These asymptotic lines provide boundaries that the hyperbola gets infinitely close to as it extends along the axes.
• They are crucial for graphing the hyperbola and understanding its orientation and spread.

By understanding the way asymptotes relate to the hyperbola's standard equation, you can more easily visualize and graph these fascinating geometric shapes.