Problem 8 - Intermediate Algebra

Question

Write the standard form of the equation of a hyperbola centered at \((h, k)\) that opens up and down.

Step-by-Step Solution

Answer

The standard form is \( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \).

Step 1: Identify Form of Hyperbola

For a hyperbola centered at \((h, k)\) that opens up and down, the standard form of the equation is \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]. Here, \(a\) and \(b\) are real numbers such that \(a^2\) and \(b^2\) represent the squares of the distances from the center to the vertices and co-vertices respectively.

Step 2: Substitute the Center

Since the problem specifies a hyperbola centered at \((h, k)\), substitute \(h\) and \(k\) directly into the equation format. The equation becomes \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \].

Step 3: Explain Components

The numerator \((y - k)^2\) shows the vertical opening by centering around \(y = k\). The \(a^2\) under \((y - k)^2\) determines the length from the center to each vertex along the y-axis. Similarly, \((x - h)^2\) with \(b^2\) accounts for the x-axis distance to co-vertices, reflecting the transverse direction.

Step 4: Final Equation Form

The hyperbola opens along the y-axis (up and down) as indicated by \((y - k)^2\) appearing first and positive. Thus, the equation is correctly stablished as \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \].

Key Concepts

Standard Form

When dealing with hyperbolas, the standard form of the equation is crucial for understanding their geometric properties. For a hyperbola that opens up and down, like the one given in the problem, the standard form is \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]. In this form, the expression \((y - k)^2\) is the leading component, reflecting vertical orientation.

• \(a^2\) represents the vertical distance, squared, from the center to each vertex.
• \(b^2\) denotes the horizontal distance, squared, from the center to each co-vertex.

Remember, the positive and negative signs indicate the difference in direction, which helps define the hyperbola's shape. This format specifically relates to vertical hyperbolas since the \(y\)-term comes first. The equality with 1 sets the scale, ensuring the hyperbola is properly sized geometrically.
It's important for keeping the hyperbola centered and correctly sized.

Center of Hyperbola

The center of a hyperbola is akin to the core or middle point from where its shape stretches out. In our equation, the center is defined by the point \((h, k)\). This center point is not arbitrary—it's critical because:

• The coordinates \(h\) and \(k\) determine the position of the hyperbola on the graph.
• From this central point, the parts of the hyperbola expand upward and downward.

The position provided by \((h, k)\) serves as a reference for the entire hyperbola's position and orientation. Being the focal point, each vertex is equidistant from this central point along the y-axis, showing its symmetry. This precise positioning helps in plotting and understanding the graph's overall layout.

Opening of Hyperbola

The direction in which a hyperbola opens provides valuable information on its orientation in the coordinate plane. For the standard form equation \( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \), this hyperbola opens vertically—up and down. Here's how we can tell:

• Since \((y - k)^2\) is the first term, the primary stretch occurs along the y-axis, revealing a vertical opening.
• The minus sign before the \((x - h)^2\) indicates that the x-direction contributes less to the shape's spread.

A vertical hyperbola resembles two mirrored 'C' shapes aligned along the y-axis. Such orientation allows you to predict symmetry and plot the hyperbola correctly on the graph. Knowing the opening gives insight into how its curve spreads, helping in understanding the geometric property.

Distance to Vertices

The concept of vertices in a hyperbola involves understanding the stretch and expansion from its center. In our vertical hyperbola, these vertices are crucial because they illustrate the spread along the y-axis.

• The vertices are found \(a\) units up and down from the center \((h, k)\).
• The length from the center to a vertex is represented by \(a\), making the total distance between vertices \(2a\).

The parameter \(a\) showcases how far these extreme points lie from the center and thereby influences the hyperbola's overall stretch. By knowing \(a\), you can plot the points exactly, marking the hyperbola’s widest spread. This metric is vital as it informs you about the overall expanse of the hyperbola in the direction it opens. Placing them correctly ensures an accurate graphical representation.