Problem 32 - Precalculus with Limits

Question

In Exercises 29-34, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. Vertices: \((0, \pm3); \quad\) asymptotes: \(y=\pm3x\)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola is \(\frac{y^2}{9} - \frac{x^2}{9} = 1\).

Step 1: Identify and define key variables

Here, 'a' is the distance from the center to a vertex, in this case is 3 (from (0, ±3)). 'b' is going to be the slope of the asymptote, which is also 3 (from y=±3x).

Step 2: Write the general equation

The general equation of a hyperbola that opens upwards and downwards is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Step 3: Substitute 'a' and 'b' into the equation

Now, substitute a and b values (which are both 3 in this case) into the general equation. So the equation becomes: \(\frac{y^2}{3^2} - \frac{x^2}{3^2} = 1\). Simplify it to get: \(\frac{y^2}{9} - \frac{x^2}{9} = 1\).

Key Concepts

Standard Form

The standard form of the hyperbola's equation helps define its shape. For a hyperbola centered at the origin with vertical transverse axis, the standard form is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). This structure allows us to identify important elements like the distances to the vertices and the slopes of the asymptotes.
When you see a hyperbola's equation, you can instantly recognize whether it opens up-down or side-to-side, depending on which term, \( y^2 \) or \( x^2 \), comes first. For vertical opening, \( y^2 \) comes first as shown, while for horizontal opening, \( x^2 \) comes first in the subtraction. Understanding the standard form is crucial for graphing a hyperbola accurately.

Vertices

Vertices are two specific points on the hyperbola that illustrate its width along the transverse axis. For our hyperbola, the vertices are at \((0, \pm3)\). Here, \(a\), the distance from the center to a vertex, is a central component.
In the format \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\):

• \(a\) is calculated as 3.
• Vertices will always be reached directly along the y-axis at \( (0, 3) \) and \( (0, -3) \) for this equation.

Recognizing the position of the vertices helps visually confirm the equation's correctness and provides insight into the dimension of the hyperbola's curves.

Asymptotes

Asymptotes are imaginary lines that a hyperbola approaches but never touches. They define the hyperbola's slant and its opening direction. In our example, the asymptotes are \(y = \pm3x\).
These lines indicate the slope \( \frac{a}{b} = \pm3 \), where:

• \( a = 3 \)
• \( b = 3 \)

The equal value of \(a\) and \(b\) shows the hyperbola is perfectly symmetrical, adding extra data to the equation's geometric representation. Always remember that asymptotes guide your drawing of the hyperbola accurately and they can be used to double-check calculations.

Equation

The core of the exercise is defining the hyperbola's equation. Plugging values into the equation of a hyperbola gives us \(\frac{y^2}{9} - \frac{x^2}{9} = 1\). This result is achieved by substituting \( a = 3 \) and \( b = 3 \) into the formula \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
Let's break down the calculation:

• Since \(a = 3\), \(a^2 = 9\).
• And because \(b = 3\), \(b^2 = 9\).

The equation represents a balanced hyperbola with identical spread in both vertical and horizontal components. Mastering the calculation and substitution into the standard form is essential for graphing and deeper algebraic problem-solving.

Center at the Origin

Placing the hyperbola at the origin makes the math cleaner and easier to handle. When the center is \((0,0)\), it simplifies the structure of the equation, avoiding additional terms.
The center position allows vertices, centers, and asymptotes to align with straightforward math:

• Vertices derive straightforwardly without added terms, lying directly on the xy-axes.
• The equation's symmetry is simplified, making transformations manageable.

Always consider the benefits of placing your hyperbola's center at the origin. It's a strategic approach that lightens the computational load and provides a cleaner graphing result.