Explain how the graph of f(x)=(-2)/(x-3)^2 can be obtained from the graph of y=1/x^2 by means of translations, compressions, expansions, or reflections.
To understand how the graph of f(x) = (-2)/(x-3)^2 can be obtained from the graph of y = 1/x^2 using translations, compressions, expansions, or reflections, we need to analyze the impact of each transformation.
1. Translation: A translation of a function involves shifting the graph horizontally or vertically. In this case, there is a horizontal translation by 3 units to the right. This means that the graph of y = 1/x^2 should be moved three units to the right to obtain the graph of f(x).
2. Expansion/Compression: An expansion or compression of a function affects the width of the graph. In this case, there is no direct expansion or compression of the quadratic function. However, the graph of y = 1/x^2 has a vertical asymptote at x = 0, while the graph of f(x) = (-2)/(x-3)^2 has a vertical asymptote at x = 3. This change in the position of the vertical asymptote is related to the compression and widening of the graph.
3. Reflection: A reflection involves flipping the graph either horizontally or vertically. In this case, there is a vertical reflection on the x-axis since the negative sign is introduced in f(x) = (-2)/(x-3)^2. This reflection causes the graph to appear upside down compared to the graph of y = 1/x^2.
In summary, to obtain the graph of f(x) = (-2)/(x-3)^2 from the graph of y = 1/x^2, we need to:
- Perform a horizontal translation of 3 units to the right.
- Apply a vertical reflection.
- Experience compression and widening of the graph due to the shift in the position of the vertical asymptote.
Note: It is always recommended to plot the graphs using technology or software to visualize the transformations accurately.