Problem 37 - College Algebra with Corequisite Support

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

The hyperbola's center is (3, 3), vertices at (3, 6) and (3, 0), and foci at approximately (3, 7.24) and (3, -1.24).

Step 1: Identify the Center

The given hyperbola equation is \(\frac{(y-3)^2}{9} - \frac{(x-3)^2}{9} = 1\). From the standard form \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), we can identify the center of the hyperbola at \((h, k)\), which in this case is \((3, 3)\).

Step 2: Determine the Vertices

For the hyperbola \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), the vertices are located at \((h, k\pm a)\). Here, \(a^2 = 9\), so \(a = 3\). Thus, the vertices are at \((3, 3\pm3)\), which means the vertices are \((3, 6)\) and \((3, 0)\).

Step 3: Calculate the Foci

The distance from the center to each focus \(c\) is determined by \(c^2 = a^2 + b^2\). Here, \(b^2 = 9\), giving \(c^2 = 9 + 9 = 18\), so \(c = \sqrt{18} = 3\sqrt{2}\). The foci are located at \((3, 3\pm3\sqrt{2})\), which are approximately \((3, 3 + 4.24) = (3, 7.24)\) and \((3, 3 - 4.24) = (3, -1.24)\).

Step 4: Sketch the Graph

To sketch the graph, start by plotting the center at \((3, 3)\). Next, plot the vertices at \((3, 6)\) and \((3, 0)\). Draw the transverse axis vertically through these points. The foci, \((3, 7.24)\) and \((3, -1.24)\), should also be plotted but not connected in the same manner. Create a rough sketch around these guides representing the shape of the hyperbola, ensuring it curves outward from the transverse axis.

Key Concepts

Understanding Conic Sections

Conic sections are fascinating geometrical shapes that arise when a plane intersects with a double-napped cone. These shapes can be identified as circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations which define it.
For hyperbolas, they are formed when the plane cuts through both nappes of the cone. This results in two distinct, mirrored curves that open away from each other.

• Hyperbolas have two branches.
• Their equation can differ based on orientation, but a common form is \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\).
• Unlike ellipses, they have a hyper-real number that models distances for their unique attributes.

By understanding conic sections, particularly hyperbolas, students can better grasp the relationship between algebraic equations and geometric figures.

Graphing the Hyperbola

Graphing a hyperbola involves understanding its equation and identifying key components like the center, axis, and asymptotes. The equation \(\frac{(y-3)^2}{9} - \frac{(x-3)^2}{9} = 1\) represents a vertical hyperbola due to the positive \(y\) term.

• The center of the hyperbola is the point \(\left(h, k\right)\). For this equation, the center is at \(\left(3, 3\right)\).
• Vertices are found from the term with positive coefficient, placed vertically or horizontally depending on orientation.
• The transverse axis shows the direction the hyperbola opens, vertically for this exercise.

When graphing:- Start by plotting the center.- Identify and draw the vertices.- Use the vertices to sketch the hyperbola's two branches.This visual representation helps in finding an intuitive understanding of how the equation dictates the shape.

Vertices and Foci

Vertices and foci are critical points that describe a hyperbola's shape and location, contributing to its geometric properties.

• Vertices are the points where each branch of the hyperbola meet the transverse axis.
• For the equation \(\frac{(y-3)^2}{9} - \frac{(x-3)^2}{9} = 1\), the vertices occur at \(\left(3, 6\right)\) and \(\left(3, 0\right)\).
• Foci are points outside the hyperbola that help in determining its shape. They also lie along the transverse axis.
• Calculated using \(c^2 = a^2 + b^2\), the foci are at \(\left(3, 3 \pm 3\sqrt{2}\right)\).

Foci and vertices guide the hyperbola's definition, helping to sketch its correct alignment and shape on a graph. Understanding and plotting these key points can enhance your grasp on visualizing hyperbolas efficiently.