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.
y = 1/x^2
scale by 2:
y = 2/x^2
reflect in x-axis:
y = -2/x^2
shift to the right by 3:
y = -2/(x-3)^2
Thank you so much!
To obtain the graph of f(x) = (-2)/(x-3)^2 from the graph of y = 1/x^2, we can observe the different transformations that have occurred.
1. Reflection: The graph of f(x) = (-2)/(x-3)^2 is a reflection of the graph of y = 1/x^2 about the y-axis. This occurs because the negative sign in front of the function (-2) reflects the graph vertically.
2. Horizontal Translation: The graph of f(x) = (-2)/(x-3)^2 is horizontally translated 3 units to the right compared to the graph of y = 1/x^2. The translation is away from the y-axis, as the positive 3 inside the parentheses shifts the graph horizontally to the right.
3. Vertical Compression/Expansion: The graph of f(x) = (-2)/(x-3)^2 is vertically compressed or expanded compared to the graph of y = 1/x^2, depending on the value of the constant (-2). If the constant was -4, for example, it would be a greater compression. In this case, the vertical compression/expansion occurs because the constant in front of the function (-2) stretches or compresses the graph vertically. In this case, -2 compresses the graph vertically.
In summary, the graph of f(x) = (-2)/(x-3)^2 is obtained from the graph of y = 1/x^2 by reflecting it vertically, translating it horizontally 3 units to the right, and vertically compressing the graph by a factor of 2.