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

(a) Vertical shift up by 1; (b) Right shift by 1 and reflection across x-axis; (c) Manipulate into form 1-function.

Step 1: Original Function

We begin with the base function \(y = \frac{1}{x}\). This is a hyperbola centered at the origin with the x and y-axes as asymptotes.

Step 2a: Transformation for (a)

To obtain \(y = 1 - \frac{1}{x}\), start with \(y = \frac{1}{x}\). Then apply the transformation of adding 1 to the entire function, resulting in a vertical shift upwards by 1 unit.

Step 2b: Transformation for (b)

To transform \(y = \frac{1}{x}\) into \(y = -\frac{1}{x-1}\), first apply a horizontal shift to the right by 1 unit (replacing \(x\) with \(x-1\)). Follow this by a reflection across the x-axis by multiplying the entire function by -1.

Step 2c: Transformation for (c)

To achieve \(y = \frac{x}{x+1}\), understand that \(\frac{x}{x + 1}\) can be seen as \(1 - \frac{1}{x+1}\). Hence, this involves combining a horizontal shift of \(x\) by 1 unit to the left inside the denominator and then adjusting the overall structure to represent \(1 - \text{function}\).

Key Concepts

Understanding the Hyperbola

A hyperbola is a type of curve on a graph that is distinctively shaped due to its asymptotic nature. Consider the function \(y = \frac{1}{x}\). This forms a hyperbola characterized by two symmetrical curves separated by the asymptotes: the x-axis and y-axis. Whenever you encounter a hyperbola, its key features include:

• Two branches opening in opposite directions (either vertically or horizontally).
• Asymptotes, which are lines the hyperbola approaches but never touches.
• A center, around which the hyperbola is symmetric, that doesn't intersect the actual graph. For \(y = \frac{1}{x}\), the center is located at the origin \((0,0)\).

Complicated as it may initially sound, understanding the basic structure of a hyperbola aids in visualizing transformations like reflections and shifts, effectively making the graphing of such functions much simpler.

Reflection Across Axes

The concept of reflection in mathematics, particularly for functions, remains a simple yet powerful tool. In essence, reflecting a graph across an axis involves mirroring it with respect to that axis.

• Across the x-axis: To reflect a function across the x-axis, multiply the function by -1. For instance, reflecting \(y = \frac{1}{x}\) across the x-axis results in \(y = -\frac{1}{x}\). This transformation inverts the graph, flipping it upside down across the horizontal line.
• Across the y-axis: Reflecting a graph across the y-axis means replacing each \(x\) with \(-x\) in the function. However, for a hyperbola like \(y = \frac{1}{x}\), this transformation does nothing because the function is inherently symmetric about the origin, a property called origin symmetry.

Understanding these reflections is crucial when performing complex transformations as they alter the orientation of a graph, affecting how shifts and other operations are approached.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are straightforward transformations that reposition the graph of a function without altering its shape. This involves either moving the function up, down, left, or right on the graph.

• Vertical Shifts: Adjusting the function by adding or subtracting a constant value changes its position along the y-axis. For instance, with \(y = \frac{1}{x}\) becoming \(y = 1 - \frac{1}{x}\), we've effectively shifted the hyperbola up by 1 unit.
• Horizontal Shifts: These shifts involve modifying the input \(x\). Replacing \(x\) with \(x + a\) (a positive \(a\)) shifts the graph left, while replacing it with \(x - a\) causes a right shift. For example, \(y = \frac{1}{x-1}\) represents a shift to the right by 1 unit compared to \(y = \frac{1}{x}\).

Incorporating these shifts allows for the precise placement of the graph in the desired position on the chart, making it an essential skill for constructing and interpreting complex graphs effectively.