Part A: Two types of transformations that can be used to transform f(x) to g(x) are translation and scaling.
Translation involves moving the graph of f(x) up or down, or left or right, to get the graph of g(x). Scaling involves stretching or compressing the graph of f(x) vertically or horizontally to get the graph of g(x).
Part B: To find k for translation, you need to determine how much you need to move the graph of f(x) to get the graph of g(x). If you are moving the graph up or down, you would add or subtract a positive or negative number k respectively. If you are moving the graph left or right, you would add or subtract a positive or negative number k inside the function's argument.
For scaling, k represents the amount by which you are stretching or compressing the graph of f(x) to get the graph of g(x). If you are stretching the graph vertically, k would be greater than 1. If you are compressing the graph vertically, k would be between 0 and 1. If you are stretching the graph horizontally, k would be between 0 and 1. If you are compressing the graph horizontally, k would be greater than 1.
Part C: To write an equation for translation, you would simply add or subtract the appropriate value of k inside or outside the function's argument. For example, f(x + k) would translate the graph k units to the left, while f(x) + k would translate the graph k units up.
To write an equation for scaling, you would multiply or divide the function by the appropriate value of k. For example, kf(x) would stretch the graph vertically by a factor of k, while f(kx) would compress the graph horizontally by a factor of k.
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