Problem 50 - College Algebra with Modeling and Visualization

Question

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x-2}+1 $$

Step-by-Step Solution

Answer

Shift \(f(x) = \frac{1}{x}\) right by 2 units and up by 1 unit to get \(g(x) = \frac{1}{x-2} + 1\). Asymptotes are at \(x=2\) and \(y=1\).

Step 1: Identify the Function for Transformation

The given function is \(g(x) = \frac{1}{x-2} + 1\). This function can best be represented by the transformation of \(f(x) = \frac{1}{x}\).

Step 2: Apply Horizontal Shift

The expression \(\frac{1}{x-2}\) suggests a horizontal transformation of the graph \(f(x) = \frac{1}{x}\). Specifically, this is a shift to the right by 2 units because of the \(x-2\) inside the denominator.

Step 3: Apply Vertical Shift

The expression \(+1\) indicates a vertical shift of the graph upward by 1 unit. This affects the entire transformed function \(\frac{1}{x-2}\), resulting in \(\frac{1}{x-2} + 1\).

Step 4: Determine Asymptotes

From the transformations, the vertical asymptote, originally at \(x = 0\) for \(f(x) = \frac{1}{x}\), is now shifted to \(x = 2\). The horizontal asymptote, initially at \(y = 0\), is moved up one unit, becoming \(y = 1\).

Step 5: Write Transformed Function

From the transformations applied, \(g(x)\) can be expressed as a transformation of \(f(x)\): \(g(x) = f(x-2) + 1\) or specifically \(g(x) = \frac{1}{x-2} + 1\), showing the relation to horizontal and vertical shifts.

Key Concepts

Understanding Horizontal Shifts

In the realm of function transformations, a horizontal shift involves moving the graph of a function left or right on the Cartesian plane. This movement occurs when the input variable, usually denoted as \(x\), is modified within the function's formula. Consider the function \(f(x)=\frac{1}{x}\) and a horizontal shift represented by \(x-2\). In this case, the graph moves right by 2 units.

Here's how it works:

• If \(f(x)\) becomes \(f(x-c)\), then the graph moves right by \(c\) units.
• If \(f(x)\) becomes \(f(x+c)\), then the graph moves left by \(c\) units.

In our exercise, \(g(x) = \frac{1}{x-2}\), the expression \(x-2\) signals that each point on the graph shifts 2 units to the right.

This transformation adjusts the position of unique points and even asymptotes of the graph, affecting the way the graph appears without changing its general shape or size.

Decoding Vertical Shifts

A vertical shift occurs when you adjust the output, or the result, of a function. Unlike horizontal shifts that move the graph left or right based on changes to the input, vertical shifts adjust the graph upward or downward through direct additions or subtractions to the function's result.

Let's dissect this concept with our function \(g(x) = \frac{1}{x-2} + 1\):

• If a function is expressed as \(f(x)+k\), the graph is moved up by \(k\) units.
• Alternatively, \(f(x)-k\) would signify a move downward by \(k\) units.

In this exercise, the \(+1\) outside the fraction signifies a shift upwards by 1 unit. Every point on the graph ascends by this amount, including any horizontal asymptotes originally set along the x-axis. By understanding this transformation, you'll see that while the graph's relative look remains unchanged, its position on the y-axis shifts, potentially changing how and where it intersects with other elements on the chart.

Exploring Asymptotes in Transformations

Asymptotes are lines that the graph of a function approaches but never actually touches. They're crucial to understanding the behavior of rational functions like \(\frac{1}{x}\) and their transformations.

For the base function \(f(x) = \frac{1}{x}\), the vertical asymptote is at \(x = 0\) and the horizontal asymptote is at \(y = 0\). Let's see how the function \(g(x) = \frac{1}{x-2} + 1\) transforms these asymptotes:

• **Vertical Asymptote**: In \(g(x) = \frac{1}{x-2} + 1\), the graph’s vertical asymptote moves from \(x = 0\) to \(x = 2\). This is due to the horizontal shift caused by \(x-2\).
• **Horizontal Asymptote**: Meanwhile, the horizontal asymptote moves up from \(y = 0\) to \(y = 1\) as a result of the vertical shift indicated by \(+1\).

Understanding these transformations is key. They help visually predict the structure of the graph and determine how the function behaves as \(x\) becomes infinitely large or small, playing an essential role in sketching and analyzing graphs.