Question

Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}-y^{2}=1$$

Short answer

2. What is the orientation of the hyperbola? 3. What are the coordinates of the vertices? 4. What are the coordinates of the foci? 5. What are the equations of the asymptotes?

Step-by-step solution

Identify the center and a, b values

The given equation follows the hyperbola standard form since it is already provided: $$\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1$$ Comparing with the given equation: $$\frac{x^{2}}{4}-y^{2}=1$$, we can deduce: The center of the hyperbola \((h, k) = (0, 0)\) Values for a and b: $$a^2 = 4, \Rightarrow a = 2$$ $$b^2 = 1, \Rightarrow b = 1$$

Determine the orientation and find the vertices

Since the x term in the given equation has a positive value, the hyperbola will have horizontal orientation. Vertices can be found using the formula: $$V_1 = (h+a, k) = (0+2, 0) = (2, 0)$$ $$V_2 = (h-a, k) = (0-2, 0) = (-2, 0)$$ So, the vertices are \((2, 0)\) and \((-2, 0)\).

Find the foci

To find the foci, we need to calculate the distance from the center, which is given by the formula: $$c = \sqrt{a^2 + b^2} = \sqrt{4 + 1} = \sqrt{5}$$ Since the hyperbola is horizontally oriented, the foci will be at: $$F_1 = (h+c, k) = (0+\sqrt{5}, 0) = (\sqrt{5}, 0)$$ $$F_2 = (h-c, k) = (0-\sqrt{5}, 0) = (-\sqrt{5}, 0)$$

Find the equations of the asymptotes

Asymptotes are lines containing the center and having a slope equal to \(± \frac{b}{a}\). Since the center is at \((0,0)\) and \(a = 2\), \(b = 1\), the asymptotes are: $$y = \pm \frac{1}{2}x$$

Graph the hyperbola and check with a graphing utility

Plot the center, vertices, and foci. Draw the asymptotes using the given equations. Now draw the hyperbola using the plotted points and asymptotes as a guide. To check the graph, use a graphing utility like Desmos or GeoGebra and enter the given equation. Check if the plotted points and graph match the one provided by the utility. In conclusion, the hyperbola has vertices at \((2, 0)\) and \((-2, 0)\), its foci at \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\), and asymptotes given by the equations \(y = \pm \frac{1}{2}x\).

Key concepts

Vertices of Hyperbola

In a hyperbola, the vertices are fundamental points that help to define its shape. The vertices are located on the transverse axis, which is the axis that passes through the center of the hyperbola, intersecting it at the vertices.

To determine the vertices of a hyperbola, you need to know the center and the value of \(a\), where \(a\) is the distance from the center to each vertex along the transverse axis. For the equation \( \frac{x^2}{4} - y^2 = 1 \), the center is at \((0, 0)\), and \(a = 2\). Therefore, the vertices can be calculated as follows:

• Vertex 1: \((h + a, k) = (0 + 2, 0) = (2, 0)\)
• Vertex 2: \((h - a, k) = (0 - 2, 0) = (-2, 0)\)

This means the hyperbola's vertices are located at coordinates \((2, 0)\) and \((-2, 0)\). Recognizing where these points are in relation to the center is crucial for sketching and understanding the graph of the hyperbola.

Foci of Hyperbola

The foci (plural for focus) of a hyperbola are essential in understanding its geometrical properties. These points lie on the hyperbola's transverse axis and help define how 'stretched' the hyperbola is.

To find the foci, we use the relationship: \(c = \sqrt{a^2 + b^2}\), where \(c\) is the distance from the center to each focus. For our given equation, \(a = 2\) and \(b = 1\), leading to:
\[ c = \sqrt{4 + 1} = \sqrt{5} \] The foci are calculated from the center as follows:

• Focus 1: \((h + c, k) = (0 + \sqrt{5}, 0) = (\sqrt{5}, 0)\)
• Focus 2: \((h - c, k) = (0 - \sqrt{5}, 0) = (-\sqrt{5}, 0)\)

These values indicate that the foci of the hyperbola are at \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\). Knowing the location of the foci helps further characterize the hyperbola's structures when creating its graph.

Asymptotes of Hyperbola

Asymptotes are lines that the hyperbola approaches but never intersects. They provide a guiding frame that shows how the branches of the hyperbola will spread out infinitely.

In standard form, the equations of the asymptotes for a hyperbola centered at \((h, k)\) are \(y - k = \pm \frac{b}{a}(x - h)\). Given that our hyperbola is centered at \((0, 0)\), we substitute \(a = 2\) and \(b = 1\) into the formula:

• Asymptote 1: \(y = \frac{1}{2}x\)
• Asymptote 2: \(y = -\frac{1}{2}x\)

These lines, \(y = x/2\) and \(y = -x/2\), are critical for drawing the hyperbola accurately on a graph, as they form the boundaries within which the hyperbola is symmetrically shaped. Observing these asymptotes helps to ensure that the branches of the hyperbola extend correctly in their respective directions.