Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{2}{(x-1)^{2}}$$

Step-by-Step Solution

Answer

Shifted right by 1, vertically stretched; domain is all reals except 1; range is positive reals.

Step 1: Analyze the base function

Start with the base function \(y = \frac{1}{x^2}\). This function has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\). Moreover, the graph is symmetrical with respect to the y-axis and has two branches, both in the first and second quadrants.

Step 2: Identify transformations

The given function is \(f(x) = \frac{2}{(x-1)^2}\). Notice the following transformations: **Horizontal Shift**: The expression \((x - 1)^2\) indicates a horizontal shift to the right by 1 unit, moving the vertical asymptote to \(x = 1\). **Vertical Stretch**: Multiplying by 2 stretches the graph vertically, amplifying the output values. There is no change in the horizontal asymptote.

Step 3: Sketch the graph by hand

For the sketch, use the transformed function information. Draw the vertical asymptote at \(x = 1\). The graph is similar to \(y = \frac{1}{x^2}\) but centered at \(x = 1\). The two branches will be in the right and left directions from \(x = 1\), narrowing as they approach \(x = 1\) and the x-axis without touching them.

Step 4: Check with a graphing calculator

Input the function \(f(x) = \frac{2}{(x-1)^2}\) into a graphing calculator to confirm the sketch. The graph should confirm the presence of the vertical asymptote at \(x = 1\) and stretch vertically, reflecting a steep increase and decrease in the values as \(x\) moves away from 1.

Step 5: Determine the domain and range

The domain of \(f(x) = \frac{2}{(x-1)^2}\): All real numbers except \(x = 1\), because the function is undefined where the denominator is zero. Thus, domain: \(x \in \mathbb{R}, x eq 1\). The range is \((0, \infty)\) because the output of \(f(x)\) is always positive and approaches infinity as \(x\) approaches 1 from either side, and approaches 0 as \(x\) becomes very large or very small.

Key Concepts

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never touches as the input variable grows larger or smaller. For the base function, \( y = \frac{1}{x^2} \), there is a vertical asymptote at \( x = 0 \). However, when transformed into \( f(x) = \frac{2}{(x-1)^2} \), the vertical asymptote shifts to \( x = 1 \).
This happens because the \( (x-1)^2 \) factor in the denominator forces the denominator to zero at \( x = 1 \), making the function undefined at this point.
Hence, you draw a vertical line at \( x = 1 \) when sketching the graph, indicating there is no output value for this input.

Horizontal Shift

Horizontal shifts affect the position of the graph along the x-axis. With our given function \( f(x) = \frac{2}{(x-1)^2} \), the expression \( (x-1) \) suggests a shift to the right by 1 unit.
This is because any input \( x \) now requires subtracting 1 first before evaluating the function, effectively moving the whole graph to the right.
The point \( x = 1 \) now corresponds to where the asymptote originally existed at \( x = 0 \) in the base graph \( y = \frac{1}{x^2} \). Remember, this doesn’t affect the shape of the graph, just its position horizontally.

Domain and Range

Understanding domain and range is essential for grasping the behavior of the function across a graph. For the function \( f(x) = \frac{2}{(x-1)^2} \), the domain includes all real numbers except \( x = 1 \). This exclusion is due to the vertical asymptote at \( x = 1 \) where the function becomes undefined.
Thus, the domain is written as \( x \in \mathbb{R}, x eq 1 \).
The range of the function is from 0 to infinity \( (0, \infty) \). Since the function's output is always positive and increasing toward infinity as \( x \) gets close to 1 from either direction, and it approaches zero as \( x \) moves away from 1 in both directions, the range does not include zero or any negative values.

Vertical Stretch

Vertical stretch modifies the graph's output values by a consistent factor, making it "steeper." In our function \( f(x) = \frac{2}{(x-1)^2} \), multiplying by 2 creates a vertical stretch.
This means every output of the base function \( y = \frac{1}{x^2} \) is doubled. Points on the graph move vertically away from the x-axis by a factor of 2.
In practical terms, this stretch makes the graph rise and fall more sharply as \( x \) approaches 1 or moves away, compared to the un-stretched base function. Such stretching ensures every result is larger, yet the overall pattern and shape of the graph remain intact, only visually more pronounced.