Problem 92 - Algebra and Trigonometry

Question

Transformations of \(y=1 / x^{2}\) 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 trans- forming 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

(a) Shift right by 2. (b) Divide to rewrite and move graph accordingly. (c) Neither function easily aligns via transformations from \( \frac{1}{x^2} \).

Step 1: Analyze the Transformation for r(x)

The function \( \frac{1}{(x-2)^2} \) is a horizontal shift of \( \frac{1}{x^2} \). To graph \( r(x) = \frac{1}{(x-2)^2} \), shift the graph of \( y = \frac{1}{x^2} \) to the right by 2 units. This doesn't affect the shape of the graph, only its position along the x-axis.

Step 2: Use Long Division for s(x)

Perform long division on \( \frac{2x^2 + 4x + 5}{x^2 + 2x + 1} \). When dividing, \( 2x^2 + 4x + 5 \) by \( x^2 + 2x + 1 \), we determine that \( 2x^2 + 4x + 5 \) divided by \( x^2 + 2x + 1 \) results in a quotient of 2 and a remainder of 3. Thus, the function can be rewritten as \( s(x) = 2 + \frac{3}{(x+1)^2} \).

Step 3: Analyze the Transformation for s(x)

To graph \( s(x) = 2 + \frac{3}{(x+1)^2} \), start by horizontally shifting \( \frac{1}{x^2} \) to the left by 1 unit. This modifies the expression to \( \frac{1}{(x+1)^2} \). Then, multiply by 3 to stretch it vertically. Finally, shift the entire graph upward by 2 units. The result is a transformation of \( \frac{1}{x^2} \) into the desired graph.

Step 4: Evaluate Transformation for p(x) and q(x)

Simplify both functions:\- For \( p(x) = \frac{2 - 3x^2}{(x-2)^2} \): The function \( \frac{1}{x^2} \) shifted horizontally and vertically cannot represent this form. Simplification does not revert back to \( \frac{1}{x^2} \).\- For \( q(x) = \frac{12x - 3x^2}{(x-2)^2} \): After simplification, it becomes apparent that \( q(x) \) cannot neatly map to a transformation of \( \frac{1}{x^2} \) using only shifts and stretches. Hence, neither can be neatly transformed from \( \frac{1}{x^2} \) without altering the form sufficiently.

Key Concepts

Transformations

Transformations in rational functions involve shifting, stretching, or reflecting the graph of a function to obtain the graph of another function. In this context, transformations allow us to modify the function \(y = \frac{1}{x^2}\) by altering its position and shape to achieve the graph of different but related functions. These transformations are powerful because they require no recalculation of specific points, just an understanding of how shifts, stretches, and reflections work.

You can perform transformations by:

• Shifting the graph horizontally or vertically, which does not alter its shape but changes its location on the coordinate plane.
• Stretching or compressing the graph, impacting its relative size and distance between points.
• Reflecting the graph over the axes, which mirrors the function across a line.

Understanding these basic concepts enables you to transform a simple graph into more complex forms efficiently.

Horizontal Shifts

Horizontal shifts occur when the entire graph of a function moves left or right along the x-axis. In the function \(y = \frac{1}{(x-2)^2}\), the shift results from replacing \(x\) with \(x-2\), causing the graph to move to the right by 2 units. This does not change the function's shape—only its position along the x-axis.

To apply a horizontal shift:

• Replace \(x\) in the function with \(x-c\) to shift right by \(c\) units.
• Replace \(x\) with \(x+c\) to shift left by \(c\) units.

This transformation doesn't affect the y-values directly, meaning features like asymptotes and the general curve of the graph remain unchanged.

Vertical Stretching

Vertical stretching involves enlarging the distances between points on a graph vertically, along the y-axis, while preserving the x-coordinates. For example, if a function is multiplied by a factor greater than 1, like the function \(y = 3\cdot\frac{1}{(x+1)^2}\), this causes the graph to stretch vertically.

To perform a vertical stretch:

• Multiply the function by a factor \( a \) where \( a > 1 \), which increases the steepness of the graph.

Conversely, if the function is multiplied by a number between 0 and 1, it compresses vertically. This stretch alters the height and makes the peaks and troughs more pronounced without changing their x-location.

Long Division

Long division is a method used to simplify complex rational functions, enabling easier graphing and transformation. In this exercise, the rational function \(s(x) = \frac{2x^2 + 4x + 5}{x^2 + 2x + 1}\) is simplified using long division to produce \(s(x) = 2 + \frac{3}{(x+1)^2}\).

Follow the steps for long division:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this result, then subtract from the original numerator.
• Repeat the process with the remainder until no further division is possible.

This step makes it evident that the original function can be rewritten in a simpler form to better facilitate transformations.