Problem 9 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

Vertices: (1, 0), (-1, 0); Foci: (√2, 0), (-√2, 0); Asymptotes: y=x, y=-x. Graph: Two branches on x-axis.

Step 1: Identify the Standard Form

The given equation of the hyperbola is \( x^{2} - y^{2} = 1 \). We can recognize this as a hyperbola in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) where \( a^2 = 1 \) and \( b^2 = 1 \). Thus, the values of \( a \) and \( b \) are both 1.

Step 2: Find the Vertices

The vertices of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are located \( a \) units away from the center along the x-axis since it is oriented along x. So, the vertices are at \( (\pm a, 0) = (\pm 1, 0) \). Thus, the vertices are \( (1, 0) \) and \( (-1, 0) \).

Step 3: Find the Foci

The foci of the hyperbola are calculated using \( c^2 = a^2 + b^2 \). Since \( a^2 = 1 \) and \( b^2 = 1 \), we get \( c^2 = 1 + 1 = 2 \). Thus, \( c = \sqrt{2} \). The foci are \( (\pm \sqrt{2}, 0) \) which are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

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 \( y = \pm \frac{b}{a}x \). Here \( a = 1 \) and \( b = 1 \), so the equations simplify to \( y = \pm x \). Therefore, the asymptotes are \( y = x \) and \( y = -x \).

Step 5: Sketch the Hyperbola

Begin by plotting the vertices at \( (1, 0) \) and \( (-1, 0) \), and the foci at \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \). Draw the asymptotes as straight lines with equations \( y = x \) and \( y = -x \). The hyperbola approaches these lines as it extends outward from the vertices. Sketch the curve by drawing the two distinct branches approaching the asymptotes through the vertices.

Key Concepts

Vertices of a Hyperbola

When understanding hyperbolas, it's crucial to start with their vertices as these define the shape and orientation. The vertices of a hyperbola are specific points where the hyperbola intersects its transverse axis. In the standard form of a hyperbola, \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] the orientation aligns along the x-axis, meaning the vertices are horizontally placed. Here, the value of \(a\) is crucial, and represents the distance from the center to each vertex along the x-axis. Given that both \(a^2 = 1\) and \(b^2 = 1\), the vertices for our particular equation \[x^2 - y^2 = 1 \] are found at

• \((1, 0)\)
• \((-1, 0)\)

The vertices serve as a handy starting guide for sketching the hyperbola and are fundamental in defining its bounds on a graph.

Foci of a Hyperbola

Foci are vital in understanding the structure of a hyperbola. They are two fixed points located along the transverse axis that define the locus of points forming the hyperbola. The relationship between the center, vertices, and foci forms the distinctive structure that differentiates hyperbolas from other conic sections. To find the foci, we utilize the formula:\[ c^2 = a^2 + b^2 \] where \(c\) represents the distance from the center to each focus. For the equation \[x^2 - y^2 = 1,\] we calculate

• \(a^2 = 1\)
• \(b^2 = 1\)
• \(c^2 = 1 + 1 = 2\)
• \(c = \sqrt{2}\)

Therefore, the foci are located at

• \((\sqrt{2}, 0)\)
• \((-\sqrt{2}, 0)\)

A clear understanding of the foci is instrumental in capturing the geometry of the hyperbola, especially when plotting or studying its properties.

Asymptotes of a Hyperbola

Asymptotes are straight lines that the branches of a hyperbola approach but never intersect. They give a sense of direction and boundary, without actually being a part of the hyperbola itself. These imaginary lines are essential in graphing the hyperbola accurately. For the standard form \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] the equations of the asymptotes can be derived from:

• \(y = \pm \frac{b}{a}x\)

In our specific hyperbola equation \[ x^2 - y^2 = 1, \] both \(a\) and \(b\) are equal to 1. This simplifies the asymptote equations to

• \(y = x\)
• \(y = -x\)

These equations represent two lines intersecting at the origin, guiding the curvature of each branch as it extends infinitely. Asymptotes help visualize the orientation and shape, providing an invisible framework around which the hyperbola wraps.