Problem 57 - Algebra 2

Question

What is an equation for the translation of \(y=\frac{2}{x}\) that has asymptotes at \(x=3\) and \(y=-5 ?\) A. \(y=\frac{2}{x-3}-5\) B. \(y=\frac{2}{x+3}+5\) C. \(y=\frac{2}{x+5}-3\) D. \(y=\frac{2}{x-5}+3\)

Step-by-Step Solution

Answer

The correct answer is A: \(y=\frac{2}{x-3}-5\)

Step 1: Identifying the translation for the asymptote at x=3

In the given equation \(y=\frac{2}{x}\), to move the vertical asymptote to the right by 3, we have to subtract 3 from \(x\), which gives us \(y=\frac{2}{x-3}\). This movement corresponds with a horizontal translation of 3 units to the right.

Step 2: Identifying the translation for the asymptote at y=-5

In the same equation \(y=\frac{2}{x-3}\) derived from the first step, to move the horizontal asymptote down by 5, we subtract 5 from \(y\), which gives us \(y=\frac{2}{x-3}-5\). This movement corresponds with a vertical translation of 5 units down.

Step 3: Combining both translations

After combining both translations, we get the equation for the hyperbola that was moved horizontally by 3 units to the right and vertically by 5 units down, which is \(y=\frac{2}{x-3}-5\).

Key Concepts

Vertical Translations

When we talk about vertical translations in the context of hyperbolas, we are referring to moving the entire graph up or down on the coordinate plane. This affects the horizontal asymptote of the hyperbola. In a vertical translation, each point on the graph shifts the same number of units in the vertical direction.

For the equation provided,

• Original: \(y = \frac{2}{x}\)
• Vertical shift: Translate vertical asymptote to \(y = -5\)

This shift is achieved by subtracting 5 from the entire equation. The equation becomes \(y = \frac{2}{x} - 5\). Here, the graph shifts 5 units downward, resulting in a new horizontal asymptote at \(y = -5\).

Vertical translations do not affect the steepness or direction of the hyperbola's branches. It simply repositions the graph along the y-axis.

Horizontal Translations

Horizontal translations involve moving the graph of a hyperbola left or right on the coordinate plane. This movement is connected to the vertical asymptote of the hyperbola. The function changes slightly to accommodate this shift.

In our original function:

• Original vertical asymptote: \(x = 0\)
• Horizontal shift needed: Translate the vertical asymptote to \(x = 3\)

This is accomplished by altering the expression inside the denominator. When you replace \(x\) with \(x - 3\), the asymptote moves to \(x = 3\), giving us \(y = \frac{2}{x-3}\). This change signifies a horizontal shift of 3 units to the right.

Horizontal translations modify where the hyperbola crosses the x-axis but do not alter the shape of the hyperbola's curves.

Asymptotes

Asymptotes are imaginary lines that the branches of a hyperbola approach but never actually reach. They act as boundary lines of sorts, guiding the direction of the hyperbola's infinite arms.

There are two primary types of asymptotes in the case of hyperbolas:

• Vertical Asymptote: Determines the horizontal placement of the hyperbola's branches.
• Horizontal Asymptote: Establishes the elevation or depression of the graph.

In the original function \(y = \frac{2}{x}\), the vertical asymptote is naturally at \(x = 0\) and the horizontal at \(y = 0\).

The problem in question requires moving these asymptotes to \(x = 3\) and \(y = -5\). By integrating both horizontal and vertical translations, the equation reaches its final form, \(y = \frac{2}{x-3} - 5\), altering the asymptotes accordingly.

Understanding asymptotes helps to predict the behavior of hyperbolas and is crucial for graphing them accurately.