Problem 38 - Algebra 2

Question

List the properties of a hyperbola that allow you to sketch its graph.

Step-by-Step Solution

Answer

To sketch a hyperbola, the necessary properties are: the center \((h,k)\) which is given by the terms in the equation, the vertices calculated using the length of the semi-major axis 'a', the foci calculated by \(c = \sqrt{a^2+b^2}\), the asymptotes which are the diagonals of the rectangle formed by \((h \pm a, k \pm b)\) or \((h \pm b, k \pm a)\) and the direction of opening which is determined by the position of the positive term in the equation.

Step 1: Identify Center

The center of the hyperbola is the midpoint of the line segment between the vertices or it is the point where asymptotes intersects. The formula to find it, if you have equation of hyperbola in standard form \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\), or \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\), is \((h,k)\), where 'h' and 'k' represents x-coordinate and y-coordinate of the center respectively.

Step 2: Identify Vertices and Foci

Vertices are the points on hyperbola that determine the ends of the major axis. For the hyperbola \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\), vertices are \((h \pm a, k)\), and for \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\), vertices are \((h, k \pm a)\). The foci are the points inside each curve of the hyperbola. The distance between center and each focus is 'c', calculated by \(c = \sqrt{a^2+b^2}\) where 'a' and 'b' are the lengths of semi-major and semi-minor axes respectively.

Step 3: Draw Asymptotes

Asymptotes are the invisible lines that hyperbola approaches but never crosses. To draw asymptotes, first draw a rectangle with center as midpoint, and the corners formed by \((h \pm a, k \pm b)\) or \((h \pm b, k \pm a)\), depending on the orientation of the hyperbola. Then, draw the diagonals of this rectangle which will serve as asymptotes of the hyperbola.

Step 4: Direction of Opening

The direction of opening can be determined by the equation of the hyperbola. If the positive term in the equation is on 'x', i.e., \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\), the hyperbola opens horizontally. Conversely, if the positive term is on 'y', i.e., \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\), the hyperbola opens vertically.

Key Concepts

Foci of Hyperbola

The foci of a hyperbola are crucial in understanding its shape and properties. Unlike an ellipse, the foci of a hyperbola are located outside the curve. These points play a significant role in the hyperbola's definition: the difference in distances from any point on the hyperbola to the two foci is constant.

The formula to find the distances to the foci involves the length of the semi-major axis \(a\) and the semi-minor axis \(b\). The distance from the center to each focus is denoted by \(c\), calculated as \(c = \sqrt{a^2 + b^2}\).

Once you've calculated \(c\), determining the exact locations of the foci is straightforward:

• For a hyperbola with a horizontal transverse axis \( (x-h)^2/a^2 - (y-k)^2/b^2 = 1\), the foci are located at \( (h \pm c, k)\).
• For a vertical hyperbola \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1\), the foci positions are \( (h, k \pm c)\).

Understanding where these points lie gives us insight into the hyperbola's overall layout.

Vertices of Hyperbola

The vertices of a hyperbola are pivotal as they define the extent of the major axis. They are the inner-most corners of the rectangle that helps determine the asymptotes, and they mark the points closest to the center where each branch of the hyperbola curves outward.

These vertices can be found using the equation of the hyperbola:

• For a hyperbola with a horizontal transverse axis, \( (x-h)^2/a^2 - (y-k)^2/b^2 = 1\), the vertices are \( (h \pm a, k)\).
• For a vertical hyperbola, \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1\), the vertices are located at \( (h, k \pm a)\).

In this configuration, \(a\) represents the distance from the center to a vertex along the major axis. By identifying the vertices, you get a clearer picture of the hyperbola's width and the direction in which it opens.

Asymptotes of Hyperbola

A hyperbola's asymptotes are lines that the curve gets infinitely close to but never crosses. These imaginary lines guide the shape of the hyperbola, ensuring its consistent curvature in both branches.

To find these asymptotes, a helpful technique is to draw a rectangle centered at \( (h, k)\), the hyperbola's center. From there, you can sketch the diagonals of the rectangle which become the asymptotes. These diagonals pass through the coordinate points aligned with the vertex direction.

• For a horizontal hyperbola, defined by \( (x-h)^2/a^2 - (y-k)^2/b^2 = 1\), the asymptotes equations are \( y = \frac{b}{a}(x-h) + k\) and \( y = -\frac{b}{a}(x-h) + k\).
• For a vertical hyperbola, expressed by \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1\), you use \( y = \frac{a}{b}(x-h) + k\) and \( y = -\frac{a}{b}(x-h) + k\).

Recognizing these asymptotes is essential for accurately graphing a hyperbola as they shape its opening patterns.

Center of Hyperbola

The center of a hyperbola is like the heart of the curve, around which everything else — from vertices to foci — is symmetrically arranged. It serves as the intersection point of the asymptotes and is typically indicated in the standard equations as \( (h, k)\).

Finding the center is the starting point for sketching a hyperbola. With the equations \( (x-h)^2/a^2 - (y-k)^2/b^2 = 1\) and \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1\), the center can be simply identified as \( (h, k)\).

This means that every calculation to find foci, vertices, and asymptotes begins at this point. Knowing the center allows for an understanding of the symmetry and provides a reference point for all other components of the hyperbola.