Problem 56 - College Algebra

Question

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$

Step-by-Step Solution

Answer

The function \(h(x)=\frac{1}{(x-3)^{2}}+2\) is graphed by transforming the original function \(f(x)=\frac{1}{x^{2}}\) by shifting right by 3 units and up by 2 units.

Step 1: Identification of the original function

Identify the original function. Here, the original function is \(f(x)=\frac{1}{x^{2}}\).

Step 2: Understanding the transformations

Understand how the function \(h(x)=\frac{1}{(x-3)^{2}}+2\) is transformed from the original function \(f(x)=\frac{1}{x^{2}}\). The denominator in \(h(x)\) is \((x-3)^{2}\), which means the graph of the original function \(f(x)\) is shifted to the right by 3 units (this makes sense because if x in \(f(x)\) was replaced by \((x-3)\), it would physically move the graph to the right by 3 units). Additionally, there is a '+2' outside the function which means the graph needs to be shifted up by 2 units.

Step 3: Graphing the function

On the grid, start by graphing the original function \(f(x)=\frac{1}{x^{2}}\). Then, apply the transformations. Shifting to the right is achieved by replacing every x-coordinate by \(x+3\), and shifting upwards is achieved by adding 2 to the y-coordinates.

Key Concepts

Transformations

Transformations involve changing the position or shape of the graph of a function. In our case, the base function is \(f(x)=\frac{1}{x^{2}}\). By applying certain transformations, we modify this graph to reach the target function \(h(x)=\frac{1}{(x-3)^{2}}+2\).
Here are the types of transformations involved:

• Horizontal Shift: When we replace \(x\) with \(x-3\), it results in a horizontal shift to the right by 3 units. This happens because the subtraction inside the bracket implies a move in the opposite direction to maintain the original range values for \(f(x)\).
• Vertical Shift: The \(+2\) added outside the function signifies a vertical shift upwards by 2 units. This raises every point on the graph so that the height increases by 2 for any given value of \(x\).

Understanding these transformations helps in graphing the modified function accurately.

Graphing Techniques

Graphing techniques for rational functions typically start with sketching the base graph and then applying the transformations step-by-step.
Let's break it down for the function \(h(x)=\frac{1}{(x-3)^{2}}+2\):

• Plot the Base Function: First, graph the function \(f(x)=\frac{1}{x^{2}}\). This graph is a basic hyperbola, symmetric about the y-axis and having a vertical asymptote at \(x=0\).
• Apply Horizontal Shift: Move the entire graph to the right by 3 units, altering the position of the center and vertical asymptote from \(x=0\) to \(x=3\).
• Apply Vertical Shift: Shift the graph upwards by 2 units so that every y-coordinate increases by 2, adjusting the highest point and distance from the x-axis.

Following these steps will ensure a smooth transition from the base function to the transformed graph, maintaining the mathematical characteristics like asymptotes and symmetry.

Asymptotes

Asymptotes are key features of rational functions, providing insights into the graph's behavior. They indicate lines that the graph approaches but never touches.
For the function \(f(x)=\frac{1}{x^{2}}\), the asymptotes are:

• Vertical Asymptote: At \(x=0\), because the function is undefined here (division by zero).
• Horizontal Asymptote: At \(y=0\), as the values of \(x\) approach positive or negative infinity, the function tends to zero.

When transforming to \(h(x)=\frac{1}{(x-3)^{2}}+2\), these asymptotes shift:

• Vertical asymptote moves to \(x=3\) due to the \(x-3\) in the denominator.
• Horizontal asymptote shifts to \(y=2\) owing to the \(+2\) in the function.

These shifts help us determine where the graph will approach but not intersect as \(x\) increases or decreases.

Vertical Shifts

Vertical shifts are transformations that move the graph of a function up or down along the y-axis.
In \(h(x)=\frac{1}{(x-3)^{2}}+2\), the term \(+2\) represents a vertical shift. Here's what you need to know:

• The entire graph of the base function \(f(x)=\frac{1}{x^{2}}\) shifts upward by 2 units.
• This affects the position of the horizontal asymptote from \(y=0\) to \(y=2\).
• Every point on \(f(x)\) is increased by 2 in the y-dimension, impacting the graph's overall height.

This ensures that the modified function \(h(x)\) retains the basic shape of \(f(x)\) but is elevated above the x-axis, affecting how it is positioned on the graph relative to the original function.