Final answer:
Transforming a linear function f(x) to g(x) can be achieved through either a vertical shift or a scaling transformation, which can include vertical stretches, compressions, or reflections. The value of k for each transformation is determined by comparing specific points on both graphs. The equations for these transformations are g(x) = f(x) + k for vertical shifts and g(x) = kf(x) for scaling transformations.
Step-by-step explanation:
To transform a linear function f(x) to another linear function g(x), there are typically two types of transformations that can be considered: a vertical shift and a scaling transformation (which includes vertical stretching or compressing and/or reflection over the x-axis).
Part A: Types of Transformations
A vertical shift occurs when every point on the graph of f(x) is moved up or down by the same amount. This shift can be represented by the function g(x) = f(x) + k, where k is the constant that determines the magnitude and direction of the shift.
A scaling transformation involves changing the steepness or orientation of the graph. If g(x) is a vertical stretch or compression of f(x), it takes the form g(x) = kf(x), where k is the scaling factor. If k>1, it is a stretch; if 0
Part B: Solving for k
To solve for k in each type of transformation, one needs to know specific points on the graphs of f(x) and g(x). By comparing the y-values of these points for the same x-value, one can determine the value of k.
Part C: Equation for Each Transformation
For the vertical shift transformation, if g(x) is f(x) shifted upward by k units, the equation would be g(x) = f(x) + k. Conversely, if the shift is downward, the equation becomes g(x) = f(x) - k.
For the scaling transformation, the equation would be g(x) = kf(x), where the value of k determines the nature of the scaling as described above.