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Chapter 2: Problem 55

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+1$$

Short Answer

Expert verified
The function \(h(x)=\frac{1}{(x-3)^{2}}+1\) is a transformation of the function \(f(x)=\frac{1}{x^2}\), which is shifted 3 units to the right and 1 unit up on the graph. The graph passes through the point (3,1)

Step by step solution

01

Identify the Basic Function

First, identify the basic function that serves as a 'starting point'. Here, it's \(f(x)=\frac{1}{x^{2}}\), a simple rational function that creates a vertically facing parabola on the graph.
02

Identify the Transformations

Next, identify the transformations that have been applied to the basic function. Here, the function \((x-3)^{2}\) in the denominator indicates a horizontal shift to the right by 3 units. The addition of 1 outside the rational function indicates a vertical shift upwards by 1 unit.
03

Apply the Transformations

Now, we apply the transformations to the basic function. Starting from the basic function's graph, shift the graph three units to the right, then one unit up. This would result in the graph passing through the point (3,1) instead of the origin (0,0).
04

Sketch the Transformed Function

Finally, sketch the transformed function, taking into account each of the identified transformations. The function \(h(x)=\frac{1}{(x-3)^{2}}+1\) is a vertically facing parabola, shifted to the right by 3 units and upwards by 1 unit.

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