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Chapter 2: Problem 56

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

Short Answer

Expert verified
The function \(h(x)=\frac{1}{(x-3)^{2}}+2\) is a transformation of the base function \(f(x)=\frac{1}{x^{2}}\), where the graph of the function is shifted 3 units to the right and then 2 units up. The resulting graph is a parabola with its vertex at the point (3,2).

Step by step solution

01

Identify the base function and the transformations

Here, the base function is \(f(x)=\frac{1}{x^{2}}\). This function already has the form of our desired function, needing only to undergo two transformations: a horizontal shift and a vertical shift. We just need to identify those shifts. We see that there is a '+3' inside the parentheses, which is equivalent to a horizontal shift of 3 steps to the right. There is also a '+2' outside of the fractional function, indicating a vertical shift of 2 steps upward.
02

Apply the horizontal shift

We graph the function \(f(x)=\frac{1}{(x-3)^{2}}\), which is the result of the base function after the horizontal shift. This means that every 'x' in the base function is replaced with '(x-3)'. This shifts the whole graph of \(f(x)=\frac{1}{x^{2}}\) 3 units to the right. The vertex of the graph that was originally at x=0 is now at x=3.
03

Apply the vertical shift

We finally add the '+2' outside our function to achieve the vertical shift. So now we are graphing \(h(x)=\frac{1}{(x-3)^{2}}+2\). This shifts the graph we obtained in step 2 two units upwards. The vertex that was at y=0 is now at y=2.
04

Final graph

The final graph of \(h(x)=\frac{1}{(x-3)^{2}}+2\) is a graph of function \(f(x)=\frac{1}{x^{2}}\), shifted 3 units to the right and 2 units up. It is an up-side down 'U' shape (or a parabola) with the vertex at the point (3,2), showing that the function behaves in a certain way around this point. Each point on the original graph has been correspondingly shifted as well in order to maintain the shape and orientation of the graph.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

You drive from your home to a vacation resort 600 miles away. You return on the same highway. The average velocity on the return trip is 10 miles per hour slower than the average velocity on the outgoing trip. Express the total time required to complete the round trip, \(T\), as a function of the average velocity on the outgoing trip, \(x .\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\).

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)
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