Problem 20 - College Algebra

Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$9 y^{2}-x^{2}=1$$

Step-by-Step Solution

Answer

The vertices are located at (0,3) and (0,-3). The foci are located at the distance of \( \sqrt{10} \) from the center (0,0). The equations of the asymptotes are y = 3x and y = -3x. The provided equation represents a hyperbola which opens upwards and downwards with regard to the y-axis.

Step 1: Identify the center, semi-major axis, and semi-minor axis

The standard form of a hyperbola is \( \frac{(y-h)^2}{a^2} - \frac{(x-k)^2}{b^2} = 1 \), for a hyperbola that opens upward or downward. From this, we can see that the center is at (0,0), a = \(\sqrt{9}\) = 3, and b = \(\sqrt{1}\) = 1.

Step 2: Compute the distance from the center to the foci

The distance from the center to each focus is given by \( c = \sqrt{a^2 + b^2} \), so in this case it is \( c = \sqrt{10} \)

Step 3: Find the vertices

The vertices are located at the distance of semi-major axis from the center along the y-axis. So the vertices are at the points (0,3) and (0,-3).

Step 4: Calculate the equations of the asymptotes

The asymptotes of the hyperbola are given by the equations \( y = ± \frac{a}{b}x \). Substituting for the values of 'a' and 'b' results in the asymptote equations y = ±3x.

Key Concepts

Asymptotes of Hyperbolas

Asymptotes are imaginary lines that provide a skeletal outline indicating the general shape of a hyperbola. They never intersect the hyperbola but come infinitely close to it.

Hyperbolas have two asymptotes that cross at the hyperbola's center, forming an 'X' shape that guides the curve of the hyperbola's branches. Determining the equations of these asymptotes for a given hyperbola is straightforward once you have the hyperbola's standard equation.For the hyperbola given by the equation \(9y^2 - x^2 = 1\), we can derive the asymptotes by rearranging the equation to its standard form. The resulting asymptote equations are \(y = \frac{a}{b}x\) and \(y = -\frac{a}{b}x\), which simplifies to \(y = \text{±}3x\) when the respective values for 'a' and 'b' are inserted, as shown in the solution to the textbook exercise.

Vertices of Hyperbolas

The vertices of a hyperbola are two distinct points on the hyperbola that lie closest to its center on the transverse axis. They are essential in determining the hyperbola’s shape and are especially critical when graphing the hyperbola by hand.

In the equation \(9y^2 - x^2 = 1\), the values for 'a' and 'b' are identified and used to locate the vertices. Since 'a' is associated with the y-term and the hyperbola opens up and down, the vertices are found at (0,3) and (0,-3), which are a distance of 'a' units from the center along the y-axis.

Foci of Hyperbolas

The foci (plural of focus) of a hyperbola are two points located inside each branch of the hyperbola. These points are unique as the difference in distances from any point on the hyperbola to the foci is constant. The foci are located along the transverse axis and are crucial for the precise definition of a hyperbola.

To find the foci of our hyperbola \(9y^2 - x^2 = 1\), we calculate the distance 'c' from the center using the formula \( c = \(a^2 + b^2\) \). Here, 'c' is \(\(a^2 + b^2\)\), which equals \(\sqrt{10}\) when the values for 'a' and 'b' are substituted. This gives the foci at (0, \(\sqrt{10}\)) and (0, -\(\sqrt{10}\))), indicating how far they are from the center along the y-axis.

Equations of Hyperbolas

Understanding the standard equations of hyperbolas is essential for various analyses, including graphing and locating fundamental components such as the center, asymptotes, vertices, and foci.

A hyperbola’s equation can be written in standard form as \(\frac{(y - h)^2}{a^2} - \frac{(x - k)^2}{b^2} = 1\) for vertical hyperbolas, or the terms may be reversed for horizontal ones. 'h' and 'k' represent the coordinates of the center, while 'a' and 'b' define distances from the center to the vertices and to the edges of the asymptote rectangle, respectively. In the case of \(9y^2 - x^2 = 1\), the standard form shows that the center is at the origin (0,0), and by solving for 'a' and 'b', we can find all other key points and lines associated with the hyperbola.