Problem 51 - 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 \(h(x)\) $$ g(x)=-\frac{2}{(x-1)^{2}} $$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is based on \(h(x)=\frac{1}{x^2}\), shifted right by 1, vertically stretched by 2, and reflected over the x-axis.

Step 1: Choose the Parent Function

First, we need to decide which parent function resembles our given function. The function \(g(x) = -\frac{2}{(x-1)^2}\) closely resembles the standard function \(h(x) = \frac{1}{x^2}\), so, we proceed by considering \(h(x)\) as our parent function.

Step 2: Apply Horizontal Shift

Observe that \(g(x) = -\frac{2}{(x-1)^2}\) contains \((x-1)\) in the denominator. This indicates a horizontal shift of the graph by 1 unit to the right. The asymptote, originally at \(x=0\) for \(h(x)\), shifts to \(x=1\).

Step 3: Apply Vertical Stretch and Reflection

The coefficient \(-2\) in \(g(x) = -\frac{2}{(x-1)^2}\) performs two transformations. The absolute value, \(2\), vertically stretches the graph by a factor of 2. The negative sign reflects the graph across the x-axis. This means that the graph, which originally resided in the first and second quadrants for \(h(x)\), will be reflected to the third and fourth quadrants for \(g(x)\).

Step 4: Identify Vertical Asymptotes

After the transformations, the vertical asymptote at \(x=0\) of \(h(x)\) is moved to \(x=1\), as indicated by \(x-1\) in the denominator.

Step 5: Identify Horizontal Asymptotes

The structure of \(g(x)\) indicates no horizontal asymptote. As \(x\) approaches infinity, \(g(x)\) approaches zero from the negative side due to the negative sign in \(-\frac{2}{(x-1)^2}\).

Step 6: Sketch the Graph

Now, with all transformations and asymptotes identified, we can sketch the graph. Begin by plotting the vertical asymptote at \(x=1\). Reflect the parent function \(h(x)\) over the x-axis, focusing the transformed graph at \(x=1\), and apply the vertical stretch.

Key Concepts

Horizontal Shift

When we talk about a horizontal shift in a graph transformation, we are referring to moving the entire graph along the x-axis. This type of transformation does not alter the shape of the graph, it only changes its position.

In the given exercise, the function is transformed from \( h(x) = \frac{1}{x^2} \) to \( g(x) = -\frac{2}{(x-1)^2} \). The expression \((x-1)\) in the denominator indicates a horizontal shift. This means the graph is moved 1 unit to the right. This shift affects the vertical asymptote, which originally sits at \( x=0 \) for the parent function \( h(x) \). After the transformation, it shifts to \( x=1 \).

• Original vertical asymptote: \( x=0 \)
• Shifted vertical asymptote: \( x=1 \)

Vertical Stretch and Reflection

A vertical stretch changes the steepness or height of a graph. Meanwhile, a reflection flips the graph across an axis.

In \( g(x) = -\frac{2}{(x-1)^2} \), the coefficient \(-2\) plays a dual role. The absolute value \(2\) indicates a vertical stretch by a factor of 2. This means the graph becomes taller or steeper compared to the original \( h(x) = \frac{1}{x^2} \).

The negative sign causes a reflection across the x-axis. Typically, \( h(x) \) is above the x-axis in the first and second quadrants. By reflecting, \( g(x) \) is moved below the x-axis, inhabiting the third and fourth quadrants.

• Steepness changes due to the factor \(2\)
• Graph flips due to the negative sign

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. Understanding these lines is essential for visualizing graph behavior as it extends towards infinity or negative infinity.

With the function \( g(x) = -\frac{2}{(x-1)^2} \), we deal with vertical and horizontal asymptotes.

For vertical asymptotes, changes occur due to the horizontal shift. In the function \( h(x) \), the vertical asymptote is at \( x=0 \). However, with \( g(x) \), this asymptote moves to \( x=1 \) due to the \( x-1 \) in the denominator. This represents where the function is undefined and the graph tends towards infinity.

Horizontal asymptotes reflect the behavior of the graph as \( x \) approaches large positive or negative values. Observing the terms in \( g(x) \), it can be noted that the graph has no real horizontal asymptote due to its structure. However, the function value approaches zero (from the negative side, considering the negative sign of the coefficient) as \( x \) grows larger. This implies the function dwindles towards the x-axis, without crossing it.