Problem 14 - College Algebra

Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. \(\frac{x^{2}}{16}-\frac{y^{2}}{25}-1\)

Step-by-Step Solution

Answer

The foci of the hyperbola are at (±√41,0). The equations of the asymptotes are y = ±(5/4)x. The hyperbola's vertices are at (±4, 0).

Step 1: Role of Constants a and b

In the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), a = 4 and b = 5 are the semi-major and semi-minor axis lengths respectively of the hyperbola.

Step 2: Find the value of c

c is the distance from the center to a focus and is found using the equation \(c = \sqrt{a^{2}+b^{2}}\). Substituting a = 4 and b = 5, we get \(c = \sqrt{4^{2}+5^{2}}= \sqrt{41}\). Our foci will be located at (±c , 0), so Foci = (± √41, 0).

Step 3: Formulation of Asymptote Equations

Since the x term has the positive coefficient, the equations of the asymptotes for a hyperbola in this form will be y = ±(b/a)x which gives us the equations y = ±(5/4)x.

Step 4: Graphing the Hyperbola along with its Asymptotes

The vertices are at points (±a,0) or (±4,0). Use this and the foci to plot the hyperbola. The asymptotes will be the lines passing through the origin making an angle of \(tan^{-1}(\frac{b}{a})\) or \(tan^{-1}(\frac{5}{4})\) with the x-axis. The graph of the hyperbola will approach these lines as the variable x or y tends to infinity.

Key Concepts

Vertices

In a hyperbola, the vertices are crucial points that define its shape and position. The vertices lie on the transverse axis, which is the axis intersected by the hyperbola itself.

• The standard form of the horizontal hyperbola equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
• For this equation, the vertices are at \((\pm a, 0)\).

Given the equation \( \frac{x^2}{16} - \frac{y^2}{25} = 1 \), we determine that \( a = 4 \), meaning:

• The vertices are located at \((4, 0)\) and \((-4, 0)\).

These points are essential when sketching the graph of the hyperbola as they mark the extremities of the curve along the horizontal line.

Asymptotes

Asymptotes are the lines that a hyperbola approaches but never actually touches. They serve as guiding boundaries that help in understanding the behavior and spread of the hyperbola.

• In the standard form equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equations of the asymptotes are \( y = \pm \frac{b}{a}x \).
• From our equation, this gives \( y = \pm \frac{5}{4}x \).

These lines pass through the origin and indicate the direction that the hyperbola's arms extend towards as they move away from the center. They form an 'X'-shaped guideline around which the hyperbola's branches open.

Foci

The foci are specific points that are located along the transverse axis, further from the center than the vertices. The distance to the foci is determined by a relationship involving the semi-major and semi-minor axes lengths.

• The formula for finding the distance to the foci from the center is \( c = \sqrt{a^2 + b^2} \).
• For our hyperbola, substituting \( a = 4 \) and \( b = 5 \) results in \( c = \sqrt{41} \).

Therefore, the foci of the hyperbola are located at \((\pm \sqrt{41}, 0)\). These points are important as they are used to define the hyperbola mathematically and are useful when calculating specific properties, such as eccentricity.

Semi-Major Axis

In the context of hyperbolas, the semi-major axis is one of the determining factors of its size and is usually the longer axis of the conic. However, unlike an ellipse, in a hyperbola the term 'semi-major axis' can be misleading, as the hyperbola does not enclose the axis.

• In this particular hyperbola, \( a = 4 \) and \( b = 5 \), designating \( b \) as the semi-major axis since \( b > a \).
• It describes the distance from the center to a vertex along the imaginary axis of the hyperbola.

The semi-major axis, along with the semi-minor axis, helps in defining the aspect ratio of the hyperbola, determining how "open" or "narrow" it appears when plotted on a graph. It is critical in the initial formulation and graphical representation of the hyperbola.