Problem 54 - Algebra and Trigonometry

Question

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$ h(x)=\frac{1}{x^{2}}-3 $$

Step-by-Step Solution

Answer

The function \( h(x)=\frac{1}{x^{2}}-3 \) is basically the graph of the function \( f(x)=\frac{1}{x^{2}} \) shifted vertically downwards by 3 units.

Step 1: Identify the Standard Function

The function \(h(x)=\frac{1}{x^{2}}-3\) is a variation of the standard function \(f(x)=\frac{1}{x^{2}}\). It involves a downward shift of this function.

Step 2: Identify the Transformation

The function \(h(x)\) is obtained from \(f(x)\) by subtracting 3. This is a vertical transformation and results in a downward shift by 3 units of the graph of \(f(x)\). This means each point on the graph of \(f(x)\) is moved 3 units downwards to form the graph of \(h(x)\).

Step 3: Plot the Transformation

Start with the graph of \(f(x)=\frac{1}{x^{2}}\), which is a curve with the general shape of an 'n' with its vertex at the origin (0,0). To plot \(h(x)\), shift the graph of \(f(x)\) downwards by 3 units. The graph of \(h(x)\) will look similar to that of \(f(x)\) but will be below it, with its vertex now at (0,-3).

Key Concepts

Transformation of Functions

When working with different types of functions, including rational functions, understanding the concept of transformation is essential. Transformations involve taking a parent function—a simple, basic function—and altering it using various operations to shift, stretch, compress, or reflect its graph. There are several types of transformations: translations (shifts), reflections across axes, and dilations (stretches and compressions).

For example, if we start with a basic rational function like \(f(x) = \frac{1}{x^2}\), adding or subtracting a constant results in a vertical shift, while multiplying or dividing by a constant outside the function would scale the graph vertically. Understanding how each of these transformations affects the graph of a function is key to mastering the art of graphing complex functions from simpler ones.

Vertical Shift

A vertical shift is a specific type of transformation that moves a graph up or down along the y-axis without altering its shape. When a constant is added to or subtracted from a function, each point on its graph moves up or down by that constant amount.

For instance, taking the parent function \(f(x) = \frac{1}{x^2}\) and subtracting 3, as in \(h(x) = \frac{1}{x^2} - 3\), results in a vertical shift of 3 units downward. It is important to visualize this as sliding the entire graph lower, which in practice means that every point on the graph \(y\) value is reduced by 3. This shift does not affect the x-values or the general shape of the graph—it simply relocates it along the vertical axis.

Parent Function

The term parent function refers to the simplest form of a function family that retains the essential characteristics of that family. In the context of rational functions, which are functions that involve the division of polynomials, commonly used parent functions are \(f(x) = \frac{1}{x}\) and \(f(x) = \frac{1}{x^2}\). These basic functions provide a foundation from which more complex functions are built through transformations.

A parent function is usually represented in its most basic form without any transformations. By understanding the graph and behavior of a parent function, one can predict how alterations to the function will change its graph. For example, knowing that \(f(x) = \frac{1}{x^2}\) has a characteristic 'n' shape with a vertex at the origin (0,0), allows us to anticipate how additions, subtractions, or scalings will alter the graph.

Plotting Functions

The process of plotting functions involves drawing the graph of a given function on a coordinate plane. This can be done by calculating and plotting individual points or by applying transformations to a well-known parent function's graph.

When plotting a transformed function like \(h(x) = \frac{1}{x^2} - 3\), a good strategy is to start with the parent function \(f(x) = \frac{1}{x^2}\) and then apply the transformation. This specific case requires us to plot the 'n' shaped curve of \(f(x)\) and then shift every point down by 3 units to accommodate the vertical shift. Plotting functions with precision is essential for understanding their characteristics and behavior, and for visually analyzing the effects of different transformations on the functions' graphs.