Problem 12 - Intermediate Algebra

Question

$$\begin{array}{|r|r|}\hline x & y \\\\\hline-2 & \\\\\hline 5 & \\\\\hline\end{array}$$ a. Fill in the blank: An equation of the form \(x y=k,\) where \(k \neq 0,\) has a graph that is a that does not intersect either the \(x\) -axis or the \(y\) -axis.

Step-by-Step Solution

Answer

The graph is a hyperbola.

Step 1: Understanding the Problem

The problem presents a table of values for variables \(x\) and \(y\), with two pairs of values, one of which is missing. It also mentions the form of the equation \(xy = k\), where \(k eq 0\). This hints at an inverse variation, where the product of \(x\) and \(y\) is constant.

Step 1: Fill the Table

Typically, for an equation of the form \(xy = k\), by substituting the known pair in the table (-2, y) and the value (5, y), we should be able to establish \(k\). However, without specific \(y\) values, we cannot determine exact numbers. The table appears to be present to help in conceptual understanding of \(x\) and \(y\) values.

Step 2: Identify the Graph Type

The equation \(xy = k\) represents a hyperbola, which is a type of conic section. The critical detail about this hyperbola is that it will not intercept the \(x\)-axis or \(y\)-axis if \(k eq 0\). This is because neither variable will be zero, as their product must be \(k\).

Step 3: Fill in the Blank

Based on the interpretation of the equation \(xy = k\), we recognize that the graph of this equation is a hyperbola that does not intersect either axis. Hence, the blank in the question refers to the type of graph: hyperbola.

Key Concepts

Hyperbola

A hyperbola is a special type of conic section that forms a curve with two separate components. When we say the equation of a hyperbola is in the form of \(xy = k\), with \(k eq 0\), it signifies a specific kind of hyperbolic relationship. This relationship is characterized by the points of the hyperbola never touching the axes, given that neither \(x\) nor \(y\) can be zero if their product is always \(k\).

Visualizing this, imagine two mirrored arcs curving away from each other, positioned opposite one another across the axes. These arcs will forever approach but never intersect the axes — this behavior results from maintaining a constant product between \(x\) and \(y\).

Key features to remember about hyperbolas include:

• They are open curves that escape to infinity.
• The separation of the curves makes them unique among conic sections.
• The center is the origin for the equation \(xy = k\), in the sense of symmetry.
• Asymptotes, if extended, would cross at the origin, guiding the shape of the hyperbola.

Conic Sections

Conic sections originate from slicing a double cone at different angles, producing several shapes, one of which is the hyperbola. They are fundamental in understanding many geometrical properties and relationships between different figures.

There are four major types of conic sections:

• **Circle** - when the cut is parallel to the base of the cone.
• **Ellipse** - when the cut is at an angle but does not pass through the base.
• **Parabola** - when the cut is parallel to the edge of the cone.
• **Hyperbola** - when the cut is steep enough to intersect both halves of the cone, which is what occurs in \(xy = k\).

Conic sections, including hyperbolas, have unique properties and equations that describe their shapes very precisely. Hyperbolas specifically stand out for having two branches which are symmetric with respect to the origin when described by \(xy = k\).

Constant Product of Variables

The constant product of variables refers to the relationship where any product of two variables, like \(x \cdot y = k\), maintains the same value \(k\). This describes an inverse variation between \(x\) and \(y\).

In inverse variation:

• When one variable increases, the other decreases to keep the product the same.
• This is why, with \(xy = k\), neither \(x\) nor \(y\) can be zero when \(k eq 0\).
• Graphically, this relationship will manifest as a hyperbola, continuously adjusting as either \(x\) or \(y\) changes.

This concept underpins the interaction between \(x\) and \(y\) in various scientific and mathematical scenarios, demonstrating balance and proportion in systems where one variable compensates for another.