Problem 26 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=1 / x^{2}\) by basic transformations: (a) \(y=\frac{1}{x^{2}}+1\) (b) \(y=-\frac{1}{(x+1)^{2}}\) (c) \(y=-\frac{1}{x^{2}}-2\)

Step-by-Step Solution

Answer

(a) Upward by 1 unit, (b) Left by 1 unit and reflected, (c) Reflected and downward by 2 units.

Step 1: Understanding the Original Function

The given function is \(y = \frac{1}{x^2}\), which is a reciprocal function asymptotic to the x-axis and y-axis. It opens upwards, with its vertex at the origin \((0, 0)\) if we consider only the right half due to symmetry.

Step 1: Vertical Shift for Part (a)

For \(y = \frac{1}{x^2} + 1\), add 1 to perform a vertical shift. The +1 shifts the entire graph of the function upward by 1 unit.

Step 2: Horizontal Shift and Reflection for Part (b)

For \(y = -\frac{1}{(x+1)^2}\), replace \(x\) with \(x+1\) to shift the graph to the left by 1 unit. The negative sign causes a reflection over the x-axis.

Step 3: Reflection and Vertical Shift for Part (c)

In \(y = -\frac{1}{x^2} - 2\), the negative sign outside the fraction indicates a reflection across the x-axis. Subtracting 2 moves the graph vertically downward by 2 units.

Key Concepts

Understanding Vertical Shifts

A vertical shift occurs when a constant is added or subtracted to a function's equation. This results in moving the graph of the function up or down on the coordinate plane without altering its shape.The mathematical expression for a vertical shift is as follows: if you have a function \[y = f(x)\]and add a constant \( c \), the new function becomes:\[y = f(x) + c\]

• Adding a positive constant \( c \) shifts the graph upward by \( c \) units.
• Adding a negative constant \( c \) shifts the graph downward by \(|c|\) units.

In the context of the exercise, for the function \(y = \frac{1}{x^2} + 1\), adding 1 shifts the original graph of \(y = \frac{1}{x^2}\) upward by 1 unit. This translation does not affect the symmetry of the graph or its asymptotic behavior with the x-axis and y-axis.

Exploring Horizontal Shifts

A horizontal shift involves moving a graph left or right along the x-axis. This transformation is achieved by adding or subtracting a constant from the variable \( x \) in the function's equation.The expression for a horizontal shift is illustrated by altering a function as follows:\[y = f(x - c)\]

• Replacing \(x\) with \(x - c\) shifts the graph to the right by \(c\) units.
• Replacing \(x\) with \(x + c\) shifts the graph to the left by \(c\) units.

For \(y = -\frac{1}{(x+1)^2}\), the presence of \(x+1\) in the denominator causes the graph to shift leftward by 1 unit. Despite this shift, the shape and orientation of the function remain the same after accounting for other transformations.

Delving into Reflections

A reflection is a transformation producing a mirror image of a graph across a specific axis. In function transformations, reflections commonly occur across the x-axis or y-axis.

• A reflection over the x-axis inverts the graph vertically. This transformation involves multiplying the entire function by -1, resulting in \(y = -f(x)\).
• A reflection over the y-axis occurs with horizontal inversion and is not covered directly in this exercise.

In selecting specific components from parts (b) and (c), let's consider the reflections:

• For part (b) \(y = -\frac{1}{(x+1)^2}\), the negative sign results in a reflection across the x-axis, flipping the graph vertically.
• Similarly, in part (c) \(y = -\frac{1}{x^2} - 2\), the graph experiences a reflection across the x-axis, inverting its orientation.

These transformations change the direction in which the graph opens but do not alter the asymptotes or horizontal positioning of the graph resulting from other transformations.