Problem 52 - 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

The graph of \(g(x)=\frac{1}{x^2} - 1\) is a vertical shift of \(h(x)=\frac{1}{x^2}\) downward by 1 unit.

Step 1: Identify the Parent Function

The function given is \(g(x) = \frac{1}{x^2} - 1\). Identify its parent function from the options, \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^2}\). The given equation matches \(h(x)=\frac{1}{x^2}\).

Step 2: Analyze the Transformation

Compare \(g(x) = \frac{1}{x^2} - 1\) with the parent function \(h(x) = \frac{1}{x^2}\). Notice that \(-1\) is subtracted from the parent function, which indicates a vertical shift downward by 1 unit.

Step 3: Write g(x) in Terms of h(x)

Since \(g(x)\) is obtained by shifting \(h(x)\) downward by 1 unit, you can express it as \(g(x) = h(x) - 1\). So, \(g(x) = \frac{1}{x^2} - 1\).

Step 4: Sketch the Graph and Add Asymptotes

Start with the basic graph of \(y = \frac{1}{x^2}\), which has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\). Applying the transformation, shift the graph down by one unit. Now, the new horizontal asymptote is at \(y = -1\), while the vertical asymptote remains at \(x = 0\).

Step 5: Verify the Graph Transformation Steps

Ensure that the graph of \(g(x) = \frac{1}{x^2} - 1\) reflects the vertical shift correctly: it should appear as \(y = \frac{1}{x^2}\) moved downward by 1 unit with the horizontal asymptote adjusted to \(y = -1\) while the vertical asymptote is at \(x = 0\).

Key Concepts

Vertical Shift in Graph Transformations

A vertical shift in graph transformations involves moving the entire graph of the function up or down along the y-axis. This is usually achieved by adding or subtracting a constant to/from the function. For example, in the function \( g(x) = \frac{1}{x^2} - 1 \), the \'-1\' indicates a vertical shift downward by one unit. This means every point on the graph of the parent function \( h(x) = \frac{1}{x^2} \) is moved one unit lower.
Understanding vertical shifts can help predict how the graph will look after transformation. It's important to recognize that this kind of shift does not alter the shape or orientation of the graph; it only affects the y-values. In our example, the points that were once at \( y = 0 \) now sit at \( y = -1 \).
In practice:

• Adding a constant moves the graph up.
• Subtracting a constant moves the graph down.

Understanding Asymptotes in Graphs

Asymptotes are lines that the graph of a function approaches but never touches. They play an essential role in sketching rational functions as they guide where the graph should go.
For the parent function \( h(x) = \frac{1}{x^2} \), the vertical asymptote is \( x = 0 \), which means the graph will soar to infinity or drop steeply as it approaches \( x = 0 \). Additionally, there is a horizontal asymptote at \( y = 0 \), signifying that as \( x \) moves further from zero, the graph aligns closer to \( y = 0 \).
After applying the vertical shift to derive \( g(x) = \frac{1}{x^2} - 1 \), the horizontal asymptote shifts to \( y = -1 \). The vertical asymptote remains unchanged at \( x = 0 \). When sketching a transformed graph:

• The position of vertical asymptotes often stays fixed.
• The horizontal asymptotes reposition according to vertical shifts.

Characteristics of Rational Functions

Rational functions are functions of the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \). These functions can exhibit various features like asymptotes, intercepts, and discontinuities, making them interesting for graph analysis.
In our example, \( g(x) = \frac{1}{x^2} - 1 \) is a rational function. Its characteristics are primarily determined by the behavior of \( \frac{1}{x^2} \) and the shift. Such functions usually have:

• Vertical asymptotes at points that make the denominator zero.
• Horizontal asymptotes that show the end behavior of the function.

With \( x = 0 \) as the problem in the denominator for \( \frac{1}{x^2} \), it results in a vertical asymptote at \( x=0 \). The function \( g(x) \) has a horizontal asymptote of \( y = -1 \) after subtraction. Understanding these elements allows for a better sketch and analysis of the graph behavior as it transforms.