Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(y^{2}-x^{2}=1\)

Step-by-Step Solution

Answer

Center: (0,0), Vertices: (0,1) and (0,-1), Foci: (0, \(\sqrt{2}\)) and (0, -\(\sqrt{2}\)), Asymptotes: lines \(y = \pm x\)

Step 1: Identify the center

The center of the hyperbola is at the origin, (0,0).

Step 2: Determine the values of 'a' and 'b'

For this particular hyperbola, a and b are square roots of the coefficients on \(y^{2}\) and \(x^{2}\). Since the equation is \(y^{2} - x^{2} = 1\), we can see that both coefficients are 1. The square root of 1 is 1, so a = b = 1.

Step 3: Find the vertices

The vertices are found at (0, +/- a) for a hyperbola with a vertical axis. This gives us vertices at points (0, 1) and (0, -1).

Step 4: Find the foci

The foci are found at (0, +/- c) where \(c = \sqrt{a^{2} + b^{2}}\). Since \(a = b = 1\), then \(c = \sqrt{1^2 + 1^2} = \sqrt{2}\). So the foci are at (0, \(\sqrt{2}\)) and (0, -\(\sqrt{2}\)).

Step 5: Write the equations of the asymptotes

The equations of the asymptotes for a hyperbola centered at the origin with a vertical transverse axis are \(y = \pm{b/a}x\). Because a = b = 1, the equations of the asymptotes are \(y = \pm x\).

Step 6: Sketch the graph

First draw the center at the origin. Then draw the vertices at (0, 1) and (0, -1) and the foci at (0, \(\sqrt{2}\)) and (0, -\(\sqrt{2}\)). Next, draw the asymptote lines using the equation \(y = \pm x\). Lastly, sketch the hyperbola to carefully approach but never touch or cross these asymptote lines.

Key Concepts

Hyperbola Characteristics

A hyperbola is a type of conic section that appears when a cone is sliced at an angle to its base. Its graph consists of two disconnected curves called branches, which mirror each other across a pair of intersecting lines called the axes of symmetry.

For the equation \(y^2 - x^2 = 1\), we have a standard form hyperbola with some distinct features:

• The center is at the origin, \((0, 0)\).
• The branches open up and down since the \(y^2\) term is positive.
• The distance from the center to the vertices is represented by 'a', which is 1 in this case.
• There is no horizontal displacement since the center is at the origin.

Understanding these characteristics is essential for graphing hyperbolas accurately and predicting their behavior.

Asymptotes of Hyperbolas

Asymptotes are lines that a curve approaches as it heads towards infinity but never actually touches. In the context of hyperbolas, asymptotes provide a framework that guides the shape of the curve.

For the hyperbola described by the equation \(y^2 - x^2 = 1\), the asymptotes are found using the general equations \(y = \pm \frac{b}{a}x\). Since our values for 'a' and 'b' are equal, the equations simplify to \(y = \pm x\).

Visualizing Asymptotes

To visualize them, draw straight lines through the center at a 45-degree angle to the axes. The relationship between the hyperbola and its asymptotes is such that the further away from the center, the closer the branches of the hyperbola get to the asymptotes.

Vertices and Foci of Hyperbolas

Vertices and foci are significant points on a hyperbola that determine its shape and orientation. For our example, \(y^2 - x^2 = 1\), the vertices are located where the hyperbola intersects the y-axis, at \((0, \pm a)\), which in this case are \((0, 1)\) and \((0, -1)\).

The foci (singular: focus) are points inside each branch of the hyperbola that dictate the spread of the curve. They're found using the formula \(c = \sqrt{a^2 + b^2}\), which in our problem yields positions at \((0, \sqrt{2})\) and \((0, -\sqrt{2})\).

Importance of Foci

The foci are essential in constructing the graph as they influence the curvature of each branch, ensuring that for any point on the hyperbola, the difference in distances to each focus is constant.

Graphing Utilities in Mathematics

Graphing utilities, such as graphing calculators and computer software, are indispensable in visualizing complex functions like hyperbolas. They allow students to input equations and observe the accompanying graphs, providing instant feedback and aiding in comprehension of the mathematical concepts at hand.

In our illustration, using a graphing utility to plot the hyperbola defined by \(y^2 - x^2 = 1\), with its foci and asymptotes, verifies the accuracy of the plotted points and drawn curves.

Advantages of Graphing Utilities

These tools are also beneficial for exploring transformations and effects of changing variables, which might be cumbersome to perform by hand. They serve not only as a means of verification but also as a way to enhance understanding by instantly visualizing abstract concepts.