Problem 70 - College Algebra

Question

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)

Step-by-Step Solution

Answer

The similarity between both hyperbolas is their shapes and orientations; both open sideways in the same way, with the same semi-axis lengths. The difference is the location of the center of each hyperbola; the first hyperbola's center is at the origin (0,0), whereas the second hyperbola's center is at (3,-3).

Step 1: Identify Center Points

From the equations of the hyperbolas, \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\), it can be observed that both graphs have the same denominators as (9,1) under \((x-h)^2\) and \((y-k)^2\) respectively. This indicates that they have the same direction of opening. However, in the first equation, the center is at (0,0), while for the second equation, the center is at (3,-3) which is translated 3 units to the right and 3 units down from the origin.

Step 2: Identify the Similarity

The similarity between both hyperbolas lies in their shape and orientation. Both hyperbolas open sideways because the positive term is associated with \(x^2\). Therefore, both hyperbolas open along the x-axis, and both have the same semi-transverse axis length (3) and semi-conjugate axis length (1), resulting in geometrically similar shapes.

Step 3: Identify the Difference

The main difference between these two hyperbolas is their location or center. The first hyperbola has center at the origin (0,0), whereas the center of the second hyperbola is at (3,-3), which is derived from shifting 3 units to the right and 3 units down from the origin. This changes the position of the hyperbola on the plane, yet their orientations remain the same.

Key Concepts

College Algebra

College algebra is a foundational course that delves into various advanced mathematical concepts, setting the stage for higher-level studies in mathematics and other sciences. It plays a critical role in understanding the behavior of functions and interpreting their graphical representations. In the context of conic sections, college algebra provides the tools to analyze equations and determine the features of their graphs, such as vertices, foci, and asymptotes. The study of hyperbolas, which are a type of conic section, forms an essential part of this subject. Students learn to recognize the standard form of a hyperbola's equation and to derive important characteristics from it, such as the direction of its opening and the location of its center, which are pivotal when comparing different hyperbolas.

Hyperbola Centers

The center of a hyperbola is a critical feature that can be identified from its equation. The general form for the equation of a hyperbola centered at the point \[\[\begin{align*}(h, k)\end{align*}\]\] is \[\[\begin{align*}\frac{(x-h)^{2}}{a^2}-\frac{(y-k)^{2}}{b^2}=1\end{align*}\]\] or \[\[\begin{align*}\frac{(y-k)^{2}}{b^2}-\frac{(x-h)^{2}}{a^2}=1,\end{align*}\]\] depending on whether the hyperbola opens horizontally or vertically. Understanding how to find the center is essential when analyzing changes in the graph of a hyperbola. For example, if the equation does not contain \[\[\begin{align*}(x-h)\end{align*}\]\] or \[\[\begin{align*}(y-k)\end{align*}\]\] terms, it is understood that the hyperbola is centered at the origin (0,0). A change in these values moves the center, and consequently, the entire graph of the hyperbola, to a new location on the coordinate plane, as seen in the exercise example.

Graph Orientation

Graph orientation refers to the direction in which a hyperbola opens on a coordinate plane. The orientation is determined by the signs and positions of the terms in the hyperbola's equation. In standard form, if the positive term is associated with \[\[\begin{align*}x^2,\end{align*}\]\] the hyperbola opens sideways, left and right, parallel to the x-axis. If the positive term is with \[\[\begin{align*}y^2,\end{align*}\]\] it opens upwards and downwards, parallel to the y-axis. This orientation affects the hyperbola's appearance and how it interacts with other geometric elements, such as lines or other conic sections. Identifying graph orientation is particularly useful when comparing hyperbolas, as it emphasizes that despite changes in location, the fundamental shape determined by orientation remains consistent.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The four primary shapes classified as conic sections are circles, ellipses, parabolas, and hyperbolas. Each type has distinct equations and properties that can be used to graph and analyze them. The exploration of these shapes forms an integral part of college algebra and other fields of mathematics, as they appear frequently in various natural and man-made contexts. Hyperbolas, specifically, are characterized by two disconnected curves that are mirror images of each other across their asymptotes. They represent the case where the plane cuts through both naps of the cone but does not intersect the vertex, resulting in an open shape that extends to infinity in its respective directions of opening.