Problem 37 - College Algebra

Question

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. $$ \frac{(y-3)^{2}}{9}-\frac{(x-3)^{2}}{9}=1 $$

Step-by-Step Solution

Answer

Center: (3,3); Vertices: (3,6), (3,0); Foci: (3,7.24), (3,-1.24).

Step 1: Identify Hyperbola Type

Since the equation is given in the form \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), it is a vertical hyperbola. The positive term contains \( y \), indicating the hyperbola opens up and down.

Step 2: Find Center, Vertices, and a,b values

The hyperbola is centered at \((h, k)\), which is \((3, 3)\). The vertices are located at \((h, k \pm a)\). Here, \(a = 3\) and \(b = 3\) since \(a^2 = 9\) and \(b^2 = 9\). Thus, the vertices are \((3, 6)\) and \((3, 0)\).

Step 3: Calculate C and Determine Foci

The foci for a vertical hyperbola are found using the equation \( c^2 = a^2 + b^2 \). Substituting the values, we get \( c^2 = 9 + 9 = 18 \). Therefore, \( c = \sqrt{18} = 3\sqrt{2} \approx 4.24 \). The foci are \((3, 3 \pm 3\sqrt{2})\), approximately \((3, 7.24)\) and \((3, -1.24)\).

Step 4: Sketch the Hyperbola

Draw the center at \((3, 3)\). Mark the vertices at \((3, 6)\) and \((3, 0)\), and the foci at \((3, 7.24)\) and \((3, -1.24)\). Sketch the branches opening upwards and downwards, passing through the vertices.

Key Concepts

Equation of Hyperbola

A hyperbola is a fascinating curve, part of a family called conic sections, which also includes parabolas and ellipses. The standard form of a hyperbola's equation is crucial because it allows us to determine important features like its center, vertices, and orientation. For the equation \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \],this format indicates a vertical hyperbola. The key here is the positive term.

• The positive term under the fraction is \((y-k)^2\), indicating the hyperbola is centered vertically.
• \(a^2\) and \(b^2\) are the denominators representing distances from the center to the vertices and co-vertices.

Understanding which term is positive helps you quickly recognize the hyperbola's orientation.

Vertices and Foci

In a hyperbola, vertices and foci are vital points that define its shape and span. The vertices are the closest points of each branch of the hyperbola to the center. For a vertical hyperbola, the vertices are found using the expression:

• \((h, k \pm a)\)

In our exercise, this translates to vertices at \((3, 6)\) and \((3, 0)\) because of the values \(a=3\), \(h=3\), and \(k=3\).
The foci are points located further along the axis than the vertices. They are crucial in defining how 'open' or 'curved' the hyperbola is. Calculate the distance to the foci using:

• \(c^2 = a^2 + b^2\)
• Plug in and find \(c = \sqrt{18}\), which makes foci at \((3, 3\pm 3\sqrt{2})\).

The calculated foci, approximately \((3, 7.24)\) and \((3, -1.24)\), help guide the sketching of the hyperbola.

Vertical Hyperbola

A vertical hyperbola is distinguished by its orientation, which is upright or straight along the y-axis. In mathematical terms, the positive term being \((y-k)^2\) means it opens vertically. A few key points include:

• Branches open upward and downward from the center, as in our exercise where the equation is \( \frac{(y-3)^{2}}{9}-\frac{(x-3)^{2}}{9}=1 \).
• The vertices lie on an imaginary vertical line through the center \((h, k)\).
• The axes of symmetry of the vertical hyperbola are aligned vertically, reflecting symmetry about the horizontal axis at the center.

Overall, recognizing a hyperbola as vertical simplifies sketching its graph.

Center of Hyperbola

The center of a hyperbola is the midpoint between its vertices and a reference point for calculating other features like vertices and foci. It is particularly important because all other key points like vertices, foci, and asymptotes are determined relative to the center.
For the general equation \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), the center is straightforwardly identified as \((h, k)\). In our exercise, it sits comfortably at \((3, 3)\).
When sketching a hyperbola:

• Start by marking the center on your graph paper.
• Then, plot the vertices and foci relative to this center using the mayor-axis and minor-axis distances \(a\) and \(b\).

Recognizing the center gives clarity to the dimensions and placement of the hyperbola on a graph.