Problem 12 - Precalculus Mathematics for Calculus

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{x^{2}}{2}-y^{2}=1$$

Step-by-Step Solution

Answer

Vertices: \( (\sqrt{2}, 0), (-\sqrt{2}, 0) \); Foci: \( ( \sqrt{3}, 0), (-\sqrt{3}, 0) \); Asymptotes: \( y = \pm \frac{1}{\sqrt{2}}x \).

Step 1: Identify the Hyperbola Type

The given equation of the hyperbola is \( \frac{x^2}{2} - y^2 = 1 \). In this form, \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the hyperbola opens left and right around the x-axis since \( x^2 \) comes first and is positive.

Step 2: Identify 'a' and 'b' Values

From the standard hyperbola equation, we compare \( \frac{x^2}{2} - \frac{y^2}{1} = 1 \) to find \( a^2 = 2 \) and \( b^2 = 1 \). Thus, \( a = \sqrt{2} \) and \( b = 1 \).

Step 3: Find the Vertices

The vertices of a hyperbola are located at \( (\pm a, 0) \). Since \( a = \sqrt{2} \), the vertices are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

Step 4: Find the Foci

The foci are found using the equation \( c^2 = a^2 + b^2 \). Here, \( c^2 = 2 + 1 = 3 \), so \( c = \sqrt{3} \). The foci are at \( (\pm \sqrt{3}, 0) \).

Step 5: Determine the Asymptotes

The asymptotes of a hyperbola with the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by \( y = \pm \frac{b}{a}x \). With \( a = \sqrt{2} \) and \( b = 1 \), the asymptotes are \( y = \pm \frac{1}{\sqrt{2}}x \).

Step 6: Sketch the Graph

Using the vertices \( (\sqrt{2}, 0), (-\sqrt{2}, 0) \), foci \( (\sqrt{3}, 0), (-\sqrt{3}, 0) \), and asymptotes \( y = \pm \frac{1}{\sqrt{2}}x \), sketch the hyperbola opening left and right along the x-axis. Plot the vertices and foci, and draw the hyperbola approaching the asymptotes.

Key Concepts

Vertices of a Hyperbola

The vertices of a hyperbola are crucial points where the hyperbola intersects its transverse axis. The transverse axis runs along the major direction of the hyperbola. For our specific equation \( \frac{x^2}{2} - y^2 = 1 \), the transverse axis is horizontal since the \( x^2 \) term is positive and comes first. The standard form of this hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, we identified \( a^2 = 2 \), so \( a = \sqrt{2} \).

When a hyperbola opens horizontally, its vertices are at \((\pm a, 0)\). Therefore, the vertices are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \). These points signify the widest part of the hyperbola, extending outwards along the x-axis. The distance between the vertices is \( 2a \), representing the length of the transverse axis. It's important to know these vertices as they help in sketching the hyperbola accurately. Plot these on a graph as the initial anchors for your hyperbolic curve.

Foci of a Hyperbola

The foci of a hyperbola are two points located on its transverse axis. They are equidistant from the center of the hyperbola and are key to its geometric definition. For our given hyperbola equation \( \frac{x^2}{2} - y^2 = 1 \), we use the equation \( c^2 = a^2 + b^2 \) to find the distance from the center to the foci.

Here, since \( a^2 = 2 \) and \( b^2 = 1 \), we calculate \( c^2 = 2 + 1 = 3 \). Thus, \( c = \sqrt{3} \). The foci are then located at \((\pm \sqrt{3}, 0)\). These points do not lie on the hyperbola itself but play a vital role in its formation.

The foci are used to define the hyperbola in terms of the distance from any point on the hyperbola to each focus. The absolute difference of these distances is always constant and equals the length of the transverse axis. This special property of the foci helps in understanding the broader geometric nature of hyperbolas.

Asymptotes of a Hyperbola

Asymptotes of a hyperbola are lines that the hyperbola approaches but never quite reaches. They provide a general direction and shape to the hyperbola, acting as imaginary boundaries for its branches. In the equation \( \frac{x^2}{2} - y^2 = 1 \), the asymptotes are calculated using the formula \( y = \pm \frac{b}{a} x \).

With \( a = \sqrt{2} \) and \( b = 1 \), the equations for the asymptotes become \( y = \pm \frac{1}{\sqrt{2}} x \). These lines pass through the origin (0,0) and have slopes of \( \pm \frac{1}{\sqrt{2}} \). Imagine them as guidelines that the branches of the hyperbola will never touch but will approach infinitely closely.

Asymptotes help in sketching the overall form of the hyperbola by ensuring the branches mirror across these lines. While they aren't part of the hyperbola itself, they provide a useful framework for visualizing the curve. In practical applications, asymptotes can be crucial for understanding scenarios like signal limits or escape velocities, where you want to know bounds rather than precise values.