If we set a and b equal to a negative number, a=b=-2 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: (0,0) , (-1,-1) , ... ,(-4,-4). This would appear to also be a slope of one.
This should be what we expected to see if we are assuming that k is associated with the slope.
If we try one more change of k say k=-2 we get:
Some coodinates of the line are now: (-4,2) , (-3,0) , ... , (0,-6). This would suggest a slope of -2.
This furthers our point even more that k affects the slope of the line.