Problem 52 - College Algebra

Question

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{(x+1)^{2}}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{1}{(x+1)^{2}}\) is obtained from the graph of the base function \(f(x)=\frac{1}{x^{2}}\) by shifting every point on it 1 unit to the left.

Step 1: Identify the base function and the transformation

The base function that has been transformed is \(f(x)=\frac{1}{x^{2}}\) and the transformation that has occurred is a horizontal shift to the left by 1 unit.

Step 2: Understand the transformation

A horizontal shift happens when a constant is added or subtracted to the input of the function. In this case, adding 1 to the input x in the denominator causes the graph of the function to shift 1 unit to the left. This can be described by the transformation \(g(x)=f(x-h)\), where h=1 for this specific problem.

Step 3: Graph the function

First, graph the base function \(f(x)=\frac{1}{x^{2}}\). Next apply the transformation to the base function: since this is a horizontal shift to the left by 1 unit, slide each point on the graph of the base function 1 unit to the left. The y coordinates stay the same. The graph of \(g(x)=\frac{1}{(x+1)^{2}}\) is then obtained.

Key Concepts

Horizontal Shift

Understanding the concept of a horizontal shift is crucial while graphing functions, especially rational functions. Essentially, a horizontal shift moves the graph of a function to the right or left along the x-axis. This type of shift does not alter the shape of the graph; it simply repositions it. This can be thought of as moving every point of the base graph left or right by a certain number of units.

For instance, let's consider a base function described by the equation, \( f(x) = \frac{1}{x^2} \). If we introduce a horizontal shift to this function, we could get a new function like \( g(x) = \frac{1}{(x + h)^2} \), where \( h \) is the horizontal shift. If \( h > 0 \), the graph moves to the left, and if \( h < 0 \), it shifts to the right. For the given exercise, \( h = 1 \), indicating a shift one unit to the left.

To visualize this, imagine every \( x \) on the graph being replaced with \( x + 1 \). This does not change the 'height' of the graph at any point because the y-values remain constant; it simply adjusts the 'horizontal position' of each point.

Transformations of Functions

In the realm of mathematics, transformations of functions are operations that change the position, shape, or size of a graph. These include translations (shifts), reflections, expansions, and contractions. When transforming a function, it's like we're manipulating the canvas on which the base graph is drawn, rather than changing the graph itself.

To begin with, a transformation such as a horizontal or vertical shift involves adding or subtracting a constant from the input (horizontal) or the output (vertical) of the function; this slides the graph in the respective direction. The example function \( g(x) = \frac{1}{(x + 1)^2} \) showcases a horizontal shift, as mentioned previously.

In addition to shifts, we have reflections which occur across the x-axis or y-axis, causing the graph to 'mirror' itself over the given axis. Scaling transformations, including stretches and compressions, either pull the graph away from or press it towards an axis.

Learning to recognize and perform these transformations allows students to graph complex functions readily by starting with a simple 'base graph' and applying the necessary changes step by step.

Algebraic Methods in Graphing

The use of algebraic methods in graphing provides a systematic way to understand and visualize functions. Especially with rational functions, algebraic techniques help in identifying asymptotes, intercepts, and important points that define the curve's progression.

To graph a rational function using algebraic methods, we should identify the following:

• The base function or starting point.
• Transformations that will be applied (shifts, stretches, reflections).
• Any asymptotes (horizontal, vertical, oblique).
• Intercepts (where the function crosses the x-axis and y-axis).
• Behavior near the asymptotes and at infinity (end behavior).

Using these steps methodically, graphing becomes a more manageable task. For the exercise at hand, the transformation is a horizontal shift applied to the function \( f(x)=\frac{1}{x^{2}} \), resulting in \( g(x)=\frac{1}{(x+1)^{2}} \). By analyzing this function algebraically, we understand that the graph retains the characteristics of \( f(x) \), but is shifted to the left by 1 unit, as highlighted in the solution. Algebraic graphing is both detailed and precise, facilitating a deeper comprehension of a function's characteristics and how they translate into its graphical representation.