If we set a and b different but one is positive and one is negative, a=-2 and b=4 and hold k constant at 1 and set t from -2 to 2 we get a graph that appears as
Some coordinates of the line are: (-4,2) , (-3,3) , ... ,(0,6). This would appear to also be a slope of one.
This still should be what we expected to see if we are assuming that k is associated with the slope. This now would lead us to one more investigation and then we will make an assumption that k changes the slope.
Let's try one more change of k say k=2 we get:
Some coodinates of the line are now: (-4,0) , (-2,2) , ... , (0,8). This would suggest a slope of 2.
This furthers our point even more that k affects the slope of the line and we can now make this assumption.