Problem 21 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 4)\) vertices \(V(0, \pm 1)\)

Step-by-Step Solution

Answer

The hyperbola's equation is \(y^2 - \frac{x^2}{15} = 1\).

Step 1: Understanding the Structure of a Hyperbola

A hyperbola with its center at the origin and vertical transverse axis has the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where \(a\) is the distance from the center to a vertex and \(c\) is the distance from the center to a focus.

Step 2: Identify the Given Parameters

From the information, the vertices are located at \(V(0, \pm 1)\) which implies \(a = 1\). The foci are located at \(F(0, \pm 4)\), indicating \(c = 4\).

Step 3: Calculate \(b^2\) Using the Relationship Between \(a\), \(b\), and \(c\)

The relationship for hyperbolas is \(c^2 = a^2 + b^2\). Substitute the known values: \(4^2 = 1^2 + b^2\). This simplifies to \(16 = 1 + b^2\).

Step 4: Solve for \(b^2\)

Rearrange the equation to find \(b^2\): \(b^2 = 16 - 1\). Thus, \(b^2 = 15\).

Step 5: Formulate the Equation of the Hyperbola

Substitute \(a^2 = 1\) and \(b^2 = 15\) into the standard form equation of the hyperbola: \(\frac{y^2}{1} - \frac{x^2}{15} = 1\). This simplifies to \(y^2 - \frac{x^2}{15} = 1\).

Key Concepts

Foci

In a hyperbola, the foci are two distinct points located symmetrically along the transverse axis of the hyperbola. These points play a crucial role in the definition and shape of the hyperbola. The foci for the hyperbola in our exercise are given as \( F(0, \pm 4) \). This indicates that the foci are positioned at a distance of 4 units away from the center, along the y-axis, because the transverse axis is vertical.

• The foci are denoted by \( c \), where \( c \) represents the distance from the center to each focus.
• Understanding the role of the foci helps in understanding the hyperbola's shape and how it opens.

For a hyperbola with a vertical transverse axis and centered at the origin, the foci coordinates are \((0, \pm c)\). In our example, \( c = 4 \), which is derived directly from the given foci locations.

Vertices

Vertices are the points where the hyperbola intersects its transverse axis. These are the closest points on each branch of the hyperbola to its center. In the exercise we looked at, the vertices are \( V(0, \pm 1) \). This indicates that the hyperbola opens vertically since the vertices are along the y-axis.

• The distance from the center to each vertex is denoted by \( a \).
• For the given hyperbola, \( a = 1 \), as derived from \( V(0, \pm 1) \).

The vertices help define the shape and orientation of the hyperbola by indicating its "width" along the transverse axis at the center.

Transverse Axis

The transverse axis of a hyperbola is akin to its axis of symmetry, passing through the vertices and foci. For hyperbolas centered at the origin, if the transverse axis is vertical, it is aligned along the y-axis. This is precisely the case in our exercise as dictated by the positions of the foci and vertices.

• The distance between the vertices defines the full length of the transverse axis.
• Here, the transverse axis is aligned vertically because the vertices and foci lie on the y-axis.

Recognizing the orientation of the transverse axis helps in writing the equation of the hyperbola correctly in the standard form.

Equation of a Hyperbola

To write the equation of a hyperbola with its center at the origin and a vertical transverse axis, one uses the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

• For our specific hyperbola, we have calculated that \( a = 1 \) and \( b^2 = 15 \).
• Substituting these into the standard equation yields \( y^2 - \frac{x^2}{15} = 1 \).

This equation provides a complete description of the hyperbola, outlining its curvature and providing a starting point for plotting or further mathematical manipulations. Understanding how the values of \( a \), \( b^2 \), and the orientation of the transverse axis are crucial to deriving this fundamental mathematical expression accurately.