Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{(y+5)^{2}}{9}-\frac{(x-4)^{2}}{25}=1\)

Step-by-Step Solution

Answer

The hyperbola has vertices at (4, -2) and (4, -8), with foci at (4, -5±√34).

Step 1: Identify the Standard Form

The given equation is \( \frac{(y+5)^2}{9} - \frac{(x-4)^2}{25} = 1 \). This is in the form of \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), where the hyperbola is vertical since the positive term is associated with \( y \).

Step 2: Determine Center, Vertices, and Foci

From the standard form, the center \((h, k)\) is \((4, -5)\). The values \(a = 3\) and \(b = 5\) are the denominators' square roots. Vertices are 3 units up and down from the center: \((4, -5+3)=(4, -2)\) and \((4, -5-3)=(4, -8)\). The distance to the foci \(c\) is found using \(c^2 = a^2 + b^2\), so \(c = \sqrt{9 + 25} = \sqrt{34}\). Foci are \((4, -5+\sqrt{34})\) and \((4, -5-\sqrt{34})\).

Step 3: Sketch the Hyperbola

Draw the center at \((4, -5)\). Plot the vertices \((4, -2)\) and \((4, -8)\). For the transverse axis, draw a vertical line segment from \((4, -8)\) to \((4, -2)\); length 6. For the asymptotes, use the slope \(\pm \frac{b}{a} = \pm \frac{5}{3}\), draw lines through the center \((4, -5)\). Hyperbola branches approach asymptotes.

Step 4: Label the Graph

On your graph, clearly label the center \((4, -5)\), the vertices \((4, -2)\) and \((4, -8)\), draw the axes and asymptotes, and mark the foci \((4, -5+\sqrt{34})\) and \((4, -5-\sqrt{34})\), noting the distance \(\sqrt{34}\) is approximately 5.83.

Key Concepts

Vertices of Hyperbola

In the context of hyperbolas, vertices are critical points that help define the shape of these fascinating curves. When you have a hyperbola in a vertical orientation, as with the equation \( \frac{(y+5)^2}{9} - \frac{(x-4)^2}{25} = 1 \), the vertices are aligned vertically. This means they sit directly above and below the center point. The center for this hyperbola is at the coordinates \((4, -5)\).

To find the vertices, we use the value of \(a\), which represents the distance from the center to each vertex along the vertical direction. In this case, \(a = 3\). By moving 3 units up and down from the center \((4, -5)\), we arrive at the vertices at \((4, -2)\) and \((4, -8)\).

These vertices are crucial because they help frame the central "box" of the hyperbola, containing the key features of the curve. Identifying and marking these points on a graph aids in visualizing the size and orientation.

Foci of Hyperbola

The foci of a hyperbola play a vital role in the definition and behavior of the curve. They lie on the same line as the vertices but are farther from the center. For our given hyperbola, we first identified the center at \((4, -5)\). From this, we calculate the "focus points," where the hyperbola's curve has significant geometric properties.

• To find the foci, we use the formula \(c^2 = a^2 + b^2\).
• Here, \(a = 3\) and \(b = 5\), giving us \(c^2 = 9 + 25 = 34\).
• Therefore, \(c = \sqrt{34} \approx 5.83\).
• The foci are then located at \((4, -5+\sqrt{34})\) and \((4, -5-\sqrt{34})\).

These points interact uniquely with the hyperbola, reflecting its eccentric shape. The idea is that exact distances from each point on the hyperbola to both foci have a consistent relationship that marks the balance between them. Including them on a sketch provides deeper insight into how the hyperbola functions.

Graph Sketching

Sketching the graph of a hyperbola can be delightful, especially when you understand the main components: the center, vertices, foci, and the shape's general orientation. First, plot the center point from the given equation. In this case, it's at \((4, -5)\). These coordinates serve as the heart of the hyperbola and the reference point for drawing other features.

Next, draw the vertices, which in our example are at \((4, -2)\) and \((4, -8)\). These points help you see the "bounds" within which the hyperbola's branches bend.

The transverse axis runs vertically through these vertices. Then, by considering the asymptotes, which have a slope \(\pm \frac{5}{3}\), lines cutting through the center point, you can understand how the branches will stretch. These asymptotes act as invisible guides that the arms of the hyperbola approach but never cross.

• Mark the center, vertices, and asymptotes clearly on your graph.
• Label the foci, ensuring they stand out from the vertices.
• Draw smooth curves that approach but do not touch the asymptotes.

Following these steps will help render an accurate representation of the hyperbola, making it an easily understandable diagram for study and reference.