Problem 84 - Algebra and Trigonometry

Question

In Example 2 we saw that some simple rational functions can be graphed by shifting, stretching, or reflecting the graph of \(y=1 / x .\) In this exercise we consider rational functions that can be graphed by transforming the graph of \(y=1 / x^{2},\) shown on the following page. (a) Graph the function $$r(x)=\frac{1}{(x-2)^{2}}$$ by transforming the graph of \(y=1 / x^{2}\) (b) Use long division and factoring to show that the function $$s(x)=\frac{2 x^{2}+4 x+5}{x^{2}+2 x+1}$$ can be written as $$s(x)=2+\frac{3}{(x+1)^{2}}$$ Then graph \(s\) by transforming the graph of \(y=1 / x^{2}\) . (c) One of the following functions can be graphed by transforming the graph of \(y=1 / x^{2} ;\) the other cannot. Use transformations to graph the one that can be, and explain why this method doesn't work for the other one. $$p(x)=\frac{2-3 x^{2}}{x^{2}-4 x+4} \quad q(x)=\frac{12 x-3 x^{2}}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

1. Shift \(y=1/x^2\) right 2 units. 2. Divide to show \(s(x) = 2 + \frac{3}{(x+1)^2}\); shift \(y=1/x^2\) left 1 unit, up 2 units. 3. Graph \(p(x); \) \(q(x)\) has an additional term, hence can't be graphed similarly.

Step 1: Graph r(x) = 1/(x-2)²

The function \(r(x) = \frac{1}{(x-2)^2}\) is obtained from \(y = \frac{1}{x^2}\) by shifting the graph 2 units to the right. The vertical asymptote moves from \(x = 0\) to \(x = 2\), and the horizontal asymptote remains \(y = 0\).

Step 2: Rewrite s(x) using long division

For \(s(x) = \frac{2x^2+7x+5}{(x+1)^2} = \frac{2x^2+7x+5}{x^2+2x+1}\). Performing polynomial long division:
\(\frac{2x^2+7x+5}{x^2+2x+1} = 2 + \frac{3x+3}{x^2+2x+1} = 2 + \frac{3(x+1)}{(x+1)^2} = 2 + \frac{3}{x+1}\)

Step 3: Graph s(x) by transformations

Since \(s(x) = 2 + \frac{3}{x+1}\), this is obtained from \(y = \frac{1}{x}\) (not \(y = \frac{1}{x^2}\)) by: (1) shifting left 1 unit to get \(\frac{1}{x+1}\), (2) stretching vertically by factor 3 to get \(\frac{3}{x+1}\), and (3) shifting up 2 units. The vertical asymptote is \(x = -1\) and the horizontal asymptote is \(y = 2\).

Key Concepts

Graph Transformations

Graph transformations are powerful tools that allow you to graph complex functions using simpler base functions. For rational functions, transforming a graph typically involves shifting it, stretching or compressing it, and reflecting it.

A common base function in rational function graphing is \( y = \frac{1}{x^2} \). Its graph looks like two mirrored curves in the first and second quadrants, both approaching the axes asymptotically.

To transform the graph of \( y = \frac{1}{x^2} \) into the graph of a function like \( r(x) = \frac{1}{(x-2)^2} \), we employ a horizontal shift. In this case, the transformation \( (x-2) \) indicates a shift of the graph 2 units to the right.

These transformations are visually represented as deviations of the graph from the origin.

• Horizontal shifts move the graph left or right.
• Vertical shifts move the graph up or down.
• Stretches and compressions change the graph's scale.
• Reflections flip the graph across an axis.

Understanding these transformations allows you to sketch the behavior of complex rational functions using simple base functions as templates.

Vertical Asymptotes

Vertical asymptotes are critical features in the graphing of rational functions. They are vertical lines that a graph approaches but never touches or crosses.

For the base function \( y = \frac{1}{x^2} \), the vertical asymptote is at \( x = 0 \). This means that as \( x \) approaches zero, the value of \( y \) approaches infinity.

In the case of transformations, vertical asymptotes shift according to the transformation applied. For instance, in \( r(x) = \frac{1}{(x-2)^2} \), the entire graph is shifted 2 units to the right. This results in the vertical asymptote moving to \( x = 2 \).

• Vertical asymptotes occur where the denominator of the rational function is zero.
• They represent the x-values which the function cannot attain.
• These lines show the boundaries of behavior for the graph.

Recognizing vertical asymptotes helps understand the limitations and features of a rational function's graph, providing insight into where the function is undefined.

Symmetry in Graphs

Symmetry is a beautiful aspect of graphing that simplifies understanding of certain functions. A graph is symmetric when it can be mirrored about a line, such as the y-axis, or rotated around a point, such as the origin.

For the base function \( y = \frac{1}{x^2} \), symmetry about the y-axis is evident. You can fold the graph along the y-axis, and both halves will align perfectly. This stems from the fact that squaring a number—whether positive or negative—yields the same result.

• Symmetry about the y-axis occurs when \( f(-x) = f(x) \).
• Symmetry about the origin occurs when \( f(-x) = -f(x) \).
• Recognizing symmetry allows prediction of the graph's shape without needing to plot every point.

In transformed graphs, symmetry may or may not be preserved, depending on the nature of the transformation. Understanding symmetry helps in predicting how the graph behaves without extensive calculations.