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Chapter 2: Problem 53

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-4$$

Short Answer

Expert verified
The graph of rational function \(h(x) = \frac{1}{x^2} - 4\) is a vertical transformation (4 units below) of the base function \(f(x) = \frac{1}{x^2}\).

Step by step solution

01

Identify the Base Function and Transformation

Here, the base function to consider is \(f(x) = \frac{1}{x^2}\). The given function is \(h(x) = \frac{1}{x^2} - 4\), which is a result of a vertical shift down of 4 units from the graph of \(f(x) = \frac{1}{x^2}\).
02

Graph the Base Function

To understand the transformation, one needs to first plot the base function. The function \(f(x) = \frac{1}{x^2}\) is a rational function having a graph which is a symmetric U shape along the y-axis and does not intersect x-axis.
03

Apply the Transformation

The transformation involves shifting all points on the graph of base function down by 4 units. In the graph, move every point on the graph of \(f(x) = \frac{1}{x^2}\) down by a distance of 4 units. This will give the graph of the function \(h(x) = \frac{1}{x^2} - 4\).
04

Plot the Transformed Function

After moving all points from the graph of \(f(x) = \frac{1}{x^2}\) down by 4 units, plot these new points to get the graph of \(h(x) = \frac{1}{x^2} - 4\). The transformed graph retains the same overall shape as the original (base function's) graph, but it's shifted 4 units down on y-axis.

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