Problem 59 - Algebra 2

Question

Explain how to find an equation for the translation of \(y=\frac{-3}{x}\) that has asymptotes at \(x=-5\) and \(y=-13 .\)

Step-by-Step Solution

Answer

The equation for the translation of \(y=\frac{-3}{x}\) that has asymptotes at \(x=-5\) and \(y=-13\) is \(y = \frac{-3}{x+5} - 13\).

Step 1: Identifying the goal

We are trying to change the asymptotes of the function \(y = \frac{-3}{x}\). The original asymptotes for this function are at x=0 and y=0. We aim to move these to new positions at x=-5 and y=-13, respectively.

Step 2: Finding the translation in the x-direction

To move the vertical asymptote from x=0 to x=-5, we need to affect the value of x in the function. We subtract -5 from x in the function to get the new function: \(y = \frac{-3}{x+5}\). This function now has vertical asymptote at x=-5.

Step 3: Finding the translation in the y-direction

To move the horizontal asymptote from y=0 to y=-13, we need to affect the value of y in the function. We subtract -13 from the whole function to get the new function: \(y = \frac{-3}{x+5} - 13\). This function now has horizontal asymptote at y=-13.

Key Concepts

Asymptotes

Asymptotes are invisible lines that guide the behavior of a graph but are never actually touched by the graph. These lines are crucial in understanding the behavior and boundaries of a rational function.
In rational functions, we typically deal with two types of asymptotes: vertical and horizontal.

• Vertical asymptotes occur when the denominator of a fraction in the function equals zero, leading to undefined values.
• Horizontal asymptotes describe the value that the function approaches as x becomes infinitely large (positive or negative).

For the function \(y=\frac{-3}{x}\):

• The vertical asymptote is at \(x=0\), because setting the denominator to zero results in an undefined expression.
• The horizontal asymptote is at \(y=0\), revealing that as \(x\) approaches infinity, the value of \(y\) approaches zero.

By understanding and manipulating these asymptotes, we can effectively translate rational functions to new locations on the graph.

Rational Functions

Rational functions are expressions that involve a ratio of two polynomials. They take the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. These functions can produce complex behavior, particularly around their asymptotes.

• The function \(y=\frac{-3}{x}\) is a simple rational function, where \(-3\) is the numerator and \(x\) is the denominator.
• Such a function graph typically has a hyperbolic shape and is divided into parts by its vertical and horizontal asymptotes, often creating two separate branches.

When transforming rational functions, our primary goal is often to recognize how shifts impact these two defining characteristics: the asymptotes. Each transformation will adjust the position of the graph while maintaining its overall shape and behavior.

Graph Transformations

Graph transformations involve shifting or altering the original graph based on specific rules or changes introduced into the function's equation. Translating graphs of rational functions requires moving their asymptotes.

• When we shift a vertical asymptote in \(y=\frac{-3}{x}\) from \(x=0\) to \(x=-5\), we need to adjust \(x\) in the function. This happens by adding \(+5\) to \(x\), resulting in \(y=\frac{-3}{x+5}\).
• The horizontal shift for \(y\) from \(0\) to \(-13\) occurs by subtracting \(-13\) from the entire function, forming \(y=\frac{-3}{x+5}-13\).

These simple transformations allow us to move the entire graph along its axes without changing its shape. Understanding these transformations will enable us to manipulate and fully comprehend the positions of asymptotes and their meanings in graph behavior.