Problem 26 - Algebra 2

Question

Write an equation for a horizontal translation of \(y=\frac{2}{x}\) Then write an equation for a vertical translation of \(y=\frac{2}{x}\) . Identify the horizontal and vertical asymptotes of the graph of each function.

Step-by-Step Solution

Answer

Horizontal translation of \(y=\frac{2}{x}\) results in \(y=\frac{2}{x-h}\), and vertical translation results in \(y=\frac{2}{x}+k\). The vertical and horizontal asymptotes for the original function are at x=0 and y=0 respectively. After horizontal translation, the vertical asymptote shifts to x=h and with vertical translation, the horizontal asymptote shifts to y=k.

Step 1: Writing an equation for a horizontal translation

Let's perform a horizontal translation of given function to the right by 'h' units. This results in: \(y = \frac{2}{(x-h)}\). By replacing x with (x-h). For example, if we want to move the graph three units to the right, the equation will be \(y = \frac{2}{(x-3)}\).

Step 2: Writing an equation for a vertical translation

The vertical translations move the graph up or down by 'k' units. This results in: \(y = \frac{2}{x} + k\). For example, to move the graph two units upward, the equation will be: \(y = \frac{2}{x} + 2\).

Step 3: Identifying the horizontal and vertical asymptotes

The vertical asymptote of the original function \(y = \frac{2}{x}\) occurs at x=0, because the function is undefined at this point. After horizontal translation, the vertical asymptote will be at x=h. For the original function, as x goes to infinity or minus infinity, y approaches 0. Thus, y = 0 is the horizontal asymptote. After the vertical translation, this asymptote will be at \(y = k\).

Key Concepts

Horizontal Translations

When we talk about horizontal translations in functions, we are essentially moving the graph of the function left or right along the x-axis. This change is made by altering the x variable within the function's equation. For example, if we begin with the function
\( y = \frac{2}{x} \),
a horizontal translation involves replacing \( x \) with \( x-h \) where \( h \) represents the shift.

• Moving the graph to the right by \( h \) units: \( y = \frac{2}{(x-h)} \)
• Moving the graph to the left by \( h \) units: \( y = \frac{2}{(x+h)} \)

For instance, if \( h = 3 \) and we move the graph three units to the right, the function becomes \( y = \frac{2}{(x-3)} \). This alters the graph's position but retains its shape.

Vertical Translations

Vertical translations in a function change the graph's position along the y-axis by adding or subtracting a constant to the function. This effectively shifts the graph up or down without affecting its horizontal placement. Let's consider our function again,
\( y = \frac{2}{x} \).
A vertical translation is accomplished by adding a constant \( k \) to the equation:
\( y = \frac{2}{x} + k \).

• To move the graph up by \( k \) units: \( y = \frac{2}{x} + k \)
• To move the graph down by \( k \) units: \( y = \frac{2}{x} - k \)

For example, if you want to shift the graph up by 2 units, the function becomes \( y = \frac{2}{x} + 2 \). This translation affects the vertical position of the graph but keeps its overall shape intact.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. They serve as boundaries showing the behavior of a function as it moves towards infinity or negative infinity.
For the initial function, \( y = \frac{2}{x} \), two types of asymptotes are present:

• Vertical asymptote: This occurs where the function is undefined. For \( y = \frac{2}{x} \), as \( x \to 0 \), the function values become extremely large, indicating a vertical asymptote at \( x = 0 \).
• Horizontal asymptote: As \( x \to \infty \) or \( x \to -\infty \), the value of \( y \) approaches 0, which is the horizontal asymptote at \( y = 0 \).

With translations, these positions can shift. A horizontal shift moves the vertical asymptote to \( x = h \), and a vertical shift moves the horizontal asymptote to \( y = k \).

Rational Functions

Rational functions are defined as functions that can be expressed as the ratio of two polynomials. The given function \( y = \frac{2}{x} \) is a straightforward example, where the numerator is a constant polynomial and the denominator is a linear polynomial.
Rational functions are significant because they often include asymptotes, like the ones we discussed.

• The denominator should not be zero, as it results in undefined values that typically indicate vertical asymptotes.
• Rational functions may have horizontal asymptotes, which suggest the end behavior as \( x \) becomes very large or very small, typically converging to a particular value or zero.

Understanding rational functions is crucial, specifically how transformations such as translations affect the graph and asymptotes. These functions can model real-world situations where quantities are inversely proportional, providing insight into how small changes in one variable can lead to significant changes in another.