Problem 14 - Precalculus: Mathematics for Calculus

Question

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$\frac{x^{2}}{2}-y^{2}=1$$

Step-by-Step Solution

Answer

Vertices: \((\sqrt{2}, 0), (-\sqrt{2}, 0)\). Foci: \((\sqrt{3}, 0), (-\sqrt{3}, 0)\). Length of transverse axis: \(2\sqrt{2}\).

Step 1: Identify the Standard Form

The given equation is \( \frac{x^2}{2} - y^2 = 1 \). This matches the standard form of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \( a^2 = 2 \) and \( b^2 = 1 \). Thus, \( a = \sqrt{2} \) and \( b = 1 \).

Step 2: Find the Vertices

For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are \( (\pm a, 0) \). Substitute \( a = \sqrt{2} \) to get the vertices at \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

Step 3: Calculate the Foci

The foci of a hyperbola are given by \( (\pm c, 0) \), where \( c = \sqrt{a^2 + b^2} \). Substitute \( a^2 = 2 \) and \( b^2 = 1 \) to get \( c = \sqrt{3} \). Thus, the foci are at \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \).

Step 4: Determine the Asymptotes

The equations of the asymptotes for the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by \( y = \pm \frac{b}{a}x \). Substitute \( a = \sqrt{2} \) and \( b = 1 \) to get asymptotes \( y = \pm \frac{x}{\sqrt{2}} \).

Step 5: Length of Transverse Axis

The length of the transverse axis of a hyperbola is \( 2a \). With \( a = \sqrt{2} \), the length of the transverse axis is \( 2\sqrt{2} \).

Step 6: Sketch the Graph

Begin by plotting the vertices \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \) on the coordinate plane. Then, draw the asymptotes \( y = \pm \frac{x}{\sqrt{2}} \). Finally, sketch the branches of the hyperbola opening left and right, approaching but never touching the asymptotes.

Key Concepts

Vertices of a Hyperbola

In a hyperbola, vertices are crucial as they represent the points where the hyperbola intersects its transverse axis. The transverse axis is the main axis running through the center of the hyperbola and along which the hyperbola opens. These vertices help in determining the shape and position of the hyperbola on a coordinate plane.

For the specific equation \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]the vertices are found at \[(\pm a, 0) \].By comparing with our given equation \[ \frac{x^2}{2} - y^2 = 1 \]we can identify \( a^2 = 2 \) and hence \( a = \sqrt{2} \). Therefore, the vertices are located at \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

These points signify where the hyperbola most distinctly crosses its axis, forming the widest section of the curve.

Foci of a Hyperbola

The foci of a hyperbola are interior points along the transverse axis, guiding the direction of the curve. These points are essential to the nature of the hyperbola because any point on the hyperbola has a constant difference from the two foci.

To determine the foci, use the formula:\[ c = \sqrt{a^2 + b^2} \]The foci are at \( (\pm c, 0) \).For our equation\[ \frac{x^2}{2} - y^2 = 1 \]where \( a^2 = 2 \ , b^2 = 1 \), we calculate \( c \) as follows:\[ c = \sqrt{2+1} = \sqrt{3} \]Thus, the foci are located at \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \).

In the graph of a hyperbola, these points reflect the twin focal points from where each curve spreads out, creating the unique hyperbolic shape.

Asymptotes of a Hyperbola

Asymptotes of a hyperbola are lines that the curve approaches but never actually reaches. They act as invisible boundaries and guide the shape and direction of the hyperbola’s branches.

The equation for the asymptotes of a hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] is \[ y = \pm \frac{b}{a}x \].For our problem, where \( a = \sqrt{2} \) and \( b = 1 \), the asymptotes are\[ y = \pm \frac{1}{\sqrt{2} }x \].

The asymptotes create a cross-like structure in the center, with each arm acting as a road. Each branch of the hyperbola travels along these asymptotic roads, widening further apart as they extend into infinity.

Transverse Axis of a Hyperbola

The transverse axis of a hyperbola is the primary axis that goes through the hyperbola, connecting the two vertices of the curve. It’s important because it helps determine the orientation and length of the hyperbola.

In our case, the transverse axis is horizontal, as indicated by the form of our equation \[ \frac{x^2}{2} - y^2 = 1 \].The length of the transverse axis is given by the formula \[2a \].With \( a = \sqrt{2} \), the length of the transverse axis is \[ 2\sqrt{2} \].

Knowing its length and direction is instrumental not only in sketching the hyperbola but also in understanding its scale and dimensions, allowing you to visualize how the curve extends in space.

Graphing Conic Sections

Graphing hyperbolas, like other conic sections, involves understanding each component: vertices, foci, asymptotes, and axes. These features, combined, form the framework needed to draw the hyperbola accurately on a coordinate plane.

For our task, follow these simple steps:

• First, plot the vertices at \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \) to give you the endpoints of the transverse axis.
• Then, mark the foci at \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \), ensuring they're well-positioned during sketching.
• Next, draw the asymptotes \( y = \pm \frac{x}{\sqrt{2}} \), which will be pivotal in guiding the curvature of the hyperbola.
• Finally, sketch the hyperbola's branches, making sure they open towards, but never touch, the asymptotes.

Each of these steps ensures a precise and helpful representation of the hyperbola on the graph.