Question 1124940: 1. (7 points) On the axes below, sketch the graph of one possible function y = f(x) be a piece-wise function satisfying all of the following requirements. Your graph should clearly show the properties listed to receive full credit.
• The domain of f is [−4, 5].
• The range of f is (−2, 6].
• f has vertical intercept (0, 2).
• f is decreasing for−4≤x<0.
• f is increasing for0≤x<2.
• f has constant rate of change for 2≤x≤5. • give your formula for f below
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
The graph is decreasing from x=-4 to x=0, at which point the function value is 2; then the graph is increasing from x=0 to x=2, and has a constant rate of change from x=2 to x=5.
Those requirements mean that the minimum value of the function between x=-4 and x=2 is at x=0, where the function value is 2. Then, in order for the function to have a range from -2 to 6, the minimum value of the function has to be at the maximum value in the domain, because the rate of change (slope) is constant from x=2 to x=5. So the graph terminates at (5,-2).
That of course means the "constant rate of change from x=2 to x=5" is negative.
The maximum value of the function has to be either at the minimum value of the domain, or at x=2. It is simpler to make it at the minimum value of the domain; so the graph "begins" at (-4,6).
Now we can choose any function value at x=2, as long as it is greater than 2 (because the graph is increasing from (0,2) to x=2) and less than or equal to 6 (because 6 is the maximum value of the range).
It turns out a function value f(2)=4 gives "nice" equations for the pieces of the graph.
So ONE POSSIBLE solution to the problem is...
f(x) = -x+2 on [-4,0] (red);
f(x) = x+2 on [0,2] (green); and
f(x) = -2x+8 on [2,5] (blue)
Note that this solution is a continuous function; the statement of the problem does not require that. So, among the many other possible solutions to the problem are some functions that are NOT continuous. But care must be taken so that any such function covers the whole defined range from -2 to 6.
Also note that the only piece of the function that is required to be linear (have a constant rate of change) is from x=2 to x=5. So the pieces from x=-4 to x=0 and from x=0 to x=2 do not have to be linear.
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