Problem 19 - Algebra and Trigonometry

Question

use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$ 4 y^{2}-x^{2}=1 $$

Step-by-Step Solution

Answer

The graph of the given hyperbola is centered at the origin and extends along the y-axis. Its vertices are at (0, ±0.5) and foci at (0, ±sqrt(1.25)). The equations of its asymptotes are y = ±0.5x.

Step 1: Write the Equation in Standard Form

The given equation is \(4y^{2} - x^{2} = 1\). This expresses a hyperbola that opens up and down. To write it in standard form, divide every term by 1, resulting in \( \frac{y^{2}}{ \frac{1}{4} } - \frac{x^{2}}{1} = 1 \), which simplifies to \( \frac{y^{2}}{0.25} - x^{2} = 1 \).

Step 2: Identify the Values of a and b

In the standard form of the equation, the square root of the denominator under \(y^{2}\) (which is 0.25) is the 'a' value and the square root of the denominator under \(x^{2}\) (which is 1) is the 'b' value. So, in this case, a=0.5 and b=1.

Step 3: Locate Vertices and Foci

The vertices are located at \((0, ±a)\) = (0, ±0.5), given the hyperbola opens up and down. To locate the foci, use the equation \(c = \sqrt{a^{2} + b^{2}}\), which yields \(c = \sqrt{0.5^{2} + 1^{2}} = \sqrt{1.25}\). Thus, the foci are at (0, ±c) = \((0, ±\sqrt{1.25})\).

Step 4: Find the Equations of the Asymptotes

The equations for the asymptotes of a hyperbola that opens up and down are \(y = ±\frac{a}{b}x\). The 'a' value is 0.5, and 'b' is 1. Thus, the equations of the asymptotes are \(y = ±0.5x\).

Key Concepts

Vertices and Asymptotes of Hyperbola

The vertices and asymptotes are crucial parts of a hyperbola's structure. Vertices are points where the hyperbola crosses its axis of symmetry. In the equation \(4y^{2}-x^{2}=1\), the hyperbola opens up and down, which means its vertices lie on the y-axis.

To find the vertices of the hyperbola, we start by putting the equation into standard form and identify the distance \(a\) from the center (0,0) to each vertex along the y-axis. In this case, the vertices are at \(0, \pm 0.5\).

Asymptotes are lines that the hyperbola approaches but never touches, providing a skeleton for graphing. For a vertical hyperbola, the asymptotes have the equation \(y = \pm \frac{a}{b}x\). In our example, the asymptotes are \(y = \pm 0.5x\). These asymptotes form the angles for the hyperbola's 'v' shapes and guide us in drawing an accurate curve.

Standard Form of a Hyperbola

Understanding the standard form of a hyperbola is like having a map while navigating an unknown city. The standard form is \( \frac{(y-k)^{2}}{a^{2}} - \frac{(x-h)^{2}}{b^{2}} = 1 \) for a vertical hyperbola or \( \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \) for a horizontal hyperbola, where \( (h,k) \) is the center.

For \(4y^{2}-x^{2}=1\), we rewrite it as \(\frac{y^{2}}{0.25} - x^{2} = 1\) to identify 'a' and 'b'. The \(a\) value represents the distance from the center to the vertices, and the \(b\) value the distance to the asymptotes. This form gives us a clear blueprint of the hyperbola's dimensions and orientation, aiding in accurate graphing and analysis of the curve.

Locating the Foci

The foci (plural of focus) of a hyperbola are points located along the major axis, inside the curve and symmetrically about the center. They are one of the defining features of hyperbolas.

To locate 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 hyperbola, \(c = \sqrt{0.5^{2} +1^{2}}\), yielding \(c = \sqrt{1.25}\). The foci then are \(0, \pm \sqrt{1.25}\), along the y-axis for this vertical hyperbola. These points are essential for constructions involving reflectivity properties characteristic of hyperbolas.

Equations of Asymptotes

The equations of the asymptotes serve as the directional guides for a hyperbola. An asymptote is a line that a curve approaches as it heads towards infinity.

For our equation \(4y^{2}-x^{2}=1\), the hyperbola opens vertically, thus we derive the asymptotes' equations using \(y = \pm \frac{a}{b}x\). With \(a = 0.5\) and \(b = 1\), we get the equations \(y = \pm 0.5x\). These lines intersect at the hyperbola's center and extend infinitely, giving us the framework to sketch the arms of the hyperbola while ensuring they never cross the asymptotes, embodying the hyperbola's open-ended nature.