Problem 34 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(y\) -intercepts \(\pm 2\) asymptotes \(y=\pm \frac{1}{4} x\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{y^2}{4} - \frac{x^2}{64} = 1\).

Step 1: Understand the Standard Form of a Hyperbola

A hyperbola has the standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) when the transverse axis is vertical. This form is applicable because the \(y\)-intercepts are given. Here, \(2a\) represents the distance between the \(y\)-intercepts.

Step 2: Determine 'a' using the y-intercepts

The \(y\)-intercepts of the hyperbola are \(\pm 2\). This implies that \(a = 2\), resulting in \(a^2 = 4\). Thus, the equation form is \(\frac{y^2}{4} - \frac{x^2}{b^2} = 1\).

Step 3: Use Asymptote Equation to Find 'b^2'

The equations of the asymptotes are \(y = \pm \frac{1}{4}x\). For a hyperbola with vertical transverse axis, the slopes of asymptotes are \(\pm \frac{a}{b}\). Hence, \(\frac{a}{b} = \frac{1}{4}\). Solve for \(b\): \(b = 4a\). Since \(a = 2\), \(b = 8\), and \(b^2 = 64\).

Step 4: Write the Equation of the Hyperbola

Substitute \(a^2\) and \(b^2\) back into the hyperbola standard form equation. The equation is \(\frac{y^2}{4} - \frac{x^2}{64} = 1\).

Key Concepts

Standard Form of a Hyperbola

Hyperbolas are fascinating curves, and their equations can be neatly written in a standard form. When a hyperbola is aligned such that the transverse axis is vertical, its standard form appears as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This formula is tailored to suit hyperbolas where the main axis runs up and down, fitting when the \( y \)-intercepts are noted.
Each term in this equation serves a purpose:

• \( y^2 \) and \( a^2 \): This part focuses on the vertical spread, crucial because the transverse axis is vertical. Here, \( a \) is half the distance between the \( y \)-intercepts.
• \( x^2 \) and \( b^2 \): This part addresses the stretch along the horizontal.

The equation follows a clear pattern. The positive term on the vertical axis dominates as the transverse axis directs upwards or downwards.

Asymptotes of a Hyperbola

Asymptotes in a hyperbola act like invisible guide lines, showing the hyperbola's direction as they stretch to infinity. Hyperbolas without the asymptotic support would stray wildly!
When a hyperbola's transverse axis is vertical, its asymptotes are equations of lines. They follow the form \( y = \pm \frac{a}{b}x \), where the plus or minus indicates the two lines moving in opposing directions.

• In our example, asymptotes are \( y = \pm \frac{1}{4} x \).
• This inform us that \( \frac{a}{b} = \frac{1}{4} \), directing us to a helpful ratio between \( a \) and \( b \).

Understanding asymptotes not only helps in sketching the hyperbola but also aids in determining the relationship between \( a \) and \( b \), essential for forming the hyperbola equation accurately.

Center of a Hyperbola

The center of a hyperbola is the point in the middle of its two distinct branches, a crucial pivot that affects the entire shape. For simplification, hyperbolas are often expressed with the center at the origin, \( (0,0) \). In such cases, the hyperbola equation aligns in a form without additional terms that push the graph around the coordinate plane.
The central location serves as a base for symmetry:

• All other points on the hyperbola are measured against this central hub.
• Both axes of the hyperbola cross at the center.

In our exercise, placing the center at the origin supports the symmetry and allows clear calculations without added complexity. This foundational point ensures each step precisely measures the hyperbola's dimensions and direction from a known benchmark.

Transverse Axis of a Hyperbola

The transverse axis of a hyperbola is no ordinary line; it is the main axis around which the hyperbola centers its curves. The transverse axis tells you the direction where the hyperbola swells the most.
In hyperbolas with a vertical transverse axis:

• The equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) affirms the focus is on the \( y \)-axis.
• \( a \): represents half the distance between the vertices along this vertical path.

For the exercise's hyperbola, the transverse axis is vertical since it has \( y \)-intercepts. Noting the values of \( a \) allows us to draw nearer to specifying how the hyperbola curves and extends. This insight helps us grasp not only the curve’s form but also its geometric focus, contributing to a deep understanding of its graph and properties.