The slope of a line never changes from point to point.  If the slope does change, the points are non-linear.  To find the slope between two points, the vertical change is divided by the horizontal change.  The vertical change is found by subtracting the y-values, and the horizontal change is found by subtracting the x-values.  If one point is (a,b) and the other is (c,d), the slope is (d-b)/(c-a). For the points in this problem, put the x-values in order: -5,-1, 2, 3, 7.  The corresponding y-values are: 17, 1, -11, -15, -31.  This gives the points (-5, 17) (-1, 1) (2, -11) (3, -15) (7, -31).

To find out if these points form a line, the slope must be found between each point.

Slope1 = (1-17)/(-1-(-5)) = -16/4 = -4

Slope2 = (-11-1)/((2-(-1)) = -12/3 = -4

Slope3 = (-15-(-11))/(3-2) = -4/1 = -4

Slope4 = (-31-(-15))/(7-3) = -16/4 = -4

Since the slope between each of the points is equal, the function f(x) is linear with a slope of -4.