Problem 25 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=1 / x\) by basic transformations: (a) \(y=1-\frac{1}{x}\) (b) \(y=-\frac{1}{x-1}\) (c) \(y=\frac{x}{x+1}\)

Step-by-Step Solution

Answer

Functions are transformed by vertical/horizontal shifts, reflection, and changes in asymptotes.

Step 1: Understand the Parent Function

The parent function given is \(y = \frac{1}{x}\), a basic rational function that has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\). The graph of \(y = \frac{1}{x}\) is a hyperbola located in the first and third quadrants of the coordinate plane.

Step 2: Transformation for (a) \(y = 1 - \frac{1}{x}\)

The function \(y = 1 - \frac{1}{x}\) can be viewed as a vertical shift of the parent function. Specifically, it is the function \(y = \frac{1}{x}\) shifted upward by 1 unit. This is because the \(-\frac{1}{x}\) portion represents the same general shape, and the '+ 1' outside shifts each point on the graph up by 1 unit.

Step 3: Transformation for (b) \(y = -\frac{1}{x-1}\)

The function \(y = -\frac{1}{x-1}\) involves two transformations. Firstly, the \(-\frac{1}{x}\) reflects the parent function across the y-axis (negative sign inverts the output). Secondly, replacing \(x\) with \(x-1\) indicates a shift to the right by 1 unit along the x-axis. This changes the vertical asymptote from \(x=0\) to \(x=1\).

Step 4: Transformation for (c) \(y = \frac{x}{x+1}\)

The function \(y = \frac{x}{x+1}\) can be seen as modifying the parent function's behavior as \(x\) approaches large values or small values. The horizontal asymptote changes to \(y=1\) because as \(x\) becomes very large, \( \frac{x}{x+1} \approx 1 \). The form of the function is more complex than a direct transformation like translation or reflection, but it can still be understood as a manipulation of the rational parent function.

Key Concepts

Rational Functions

A rational function is a fraction in which both the numerator and the denominator are polynomials. The simplest example is the parent function given, which is \[ y = \frac{1}{x} \]. Understanding rational functions involves examining their behavior, especially what happens when they approach unknown points, like points where the denominator is zero.- Rational functions can have up to two types of asymptotes: - **Vertical Asymptotes** occur when the denominator is zero. - **Horizontal Asymptotes** depend on the relationship between the degrees of the numerator and the denominator.Rational functions often resemble hyperbolas and have many unique transformations. These transformations include shifting, reflecting, and stretching or compressing the graph. Knowing the basic form of the function helps to predict its graph efficiently.

Vertical Shifts

Vertical shifts move the graph of a function up or down in the coordinate plane. These types of transformations are represented by constants added or subtracted from the entire function. In algebraic terms, if you have a function \( y = f(x) \), adding a constant \( c \) creates a vertical shift:\[ y = f(x) + c \]. For example, the transformation of \( y = \frac{1}{x} \) to \( y = 1 - \frac{1}{x} \) represents a vertical shift. Each point on the graph of the hyperbola moves up by 1 unit, altering the horizontal asymptote to \( y = 1 \) while the shape remains unchanged. This transformation visually shifts all points upward, without changing the symmetry or orientation of the graph.

Horizontal Shifts

Horizontal shifts adjust the graph left or right on the coordinate plane. They occur when there is a change inside the function argument, altering its input values.For the function \( y = f(x) \), a horizontal shift to the right by \( c \) units can be represented as:\[ y = f(x - c) \]. Conversely, a shift left is presented as \( y = f(x + c) \). A clear example of a horizontal shift is seen in the transformation from \( y = \frac{1}{x} \) to \( y = -\frac{1}{x-1} \). Here, replacing \( x \) by \( x-1 \) shifts the graph 1 unit to the right. This adjustment alters the location of the vertical asymptote from \( x = 0 \) to \( x = 1 \), keeping other characteristics unchanged.

Reflections

Reflections in transformations flip the graph over a specified line, such as the x-axis or y-axis. The reflection can be particularly noticeable in rational functions where signs play a critical role.For example, the transformation of \( y = \frac{1}{x} \) to \( y = -\frac{1}{x-1} \) involves a reflection across the x-axis. The negative sign before the fraction indicates this transformation. Initially positioned in the first and third quadrants, reflecting the function places it into the second and fourth quadrants, effectively "inverting" the values. This flip does not change the horizontal or vertical asymptotes but does affect the orientation of the hyperbola's arms.

Asymptotes

Asymptotes are vital in defining the behavior of rational functions as they point towards infinity. They are lines that a graph approaches but never actually touches.- **Vertical Asymptotes** occur where the function’s denominator is zero. It signifies a point where the graph heads off to positive or negative infinity. For instance, \( y = \frac{1}{x} \) has a vertical asymptote at \( x = 0 \).- **Horizontal Asymptotes** focus on end behavior, determined by comparing the degrees of the numerator and denominator or, as in \( y = \frac{x}{x+1} \), analyzing the limits as \( x \) approaches infinity.Horizontal asymptotes provide insight into how the function behaves as \( x \) becomes very large or very small. In this case, \( y = \frac{x}{x+1} \) approaches the line \( y = 1 \) as \( x \to \infty \), a reflection of the balancing effect between the growth of the numerator and denominator.