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Chapter 2: Problem 54

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

Short Answer

Expert verified
The graph of the function \( h(x) = \frac{1}{x^{2}} - 3 \) is a transformation of the graph of the parent function \( f(x) = \frac{1}{x^{2}} \), specifically it is shifted downwards by 3 units. The parent function \( f(x) = \frac{1}{x^{2}} \) is a U-shape in quadrant I and a flipped U-shape in quadrant II. When it is translated downwards by 3 units, the U-shape and flipped U-shape both move downwards but retain their original shape, giving the graph for the function \( h(x) = \frac{1}{x^{2}} - 3 \).

Step by step solution

01

- Understand The Parent Function

Firstly, note that \( f(x) = \frac{1}{x^{2}} \) is the parent function to the given function. Graph \( f(x) = \frac{1}{x^{2}} \). It looks a lot like \( f(x) = \frac{1}{x} \), but with a critical difference. While the graph of \( f(x) = \frac{1}{x} \) passes through all 4 quadrants, the graph of \( f(x) = \frac{1}{x^{2}} \) only passes through quadrants I and II because the square of any real number (other than zero) is always positive. This gives a U-shaped curve in quadrant I and a flipped U-shaped curve in quadrant II, with the origin being an asymptote.
02

- Identification of Transformation Type

Now take a look at the given function \( h(x) = \frac{1}{x^{2}} - 3 \). Here we can see that it is the parent function \( f(x) = \frac{1}{x^{2}} \) subtracted by 3. This is a vertical shift transformation, specifically a vertical shift downwards by 3 units. This means the whole graph of \( f(x) = \frac{1}{x^{2}} \) will move down by 3 units to give the graph of \( h(x) = \frac{1}{x^{2}} - 3 \).
03

- Graph the Transformed Function

Next let's graph the the function \( h(x) = \frac{1}{x^{2}} - 3 \) using the understanding from the previous step. Keep the U-shape of \( f(x) = \frac{1}{x^{2}} \) but move it down by 3 units on the y-axis. You will notice that the function \( h(x) = \frac{1}{x^{2}} - 3 \) also does not pass through the x-axis because it has been translated downwards.

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