your two points are (-4,14) and (2,2)
the average rate of change between those two points would be equal to (2 - 14) / (2 - -4) = -12/6 = -2.
for every value that x increases by 1, the value of g(x) decreases by 2.
you can make a table as shown below:
x g(x) -4 14 -3 12 -2 10 -1 8 0 6 1 4 2 2
you can graph the line of the average rate of change between those two points as follows:
slope intercept form of the equation of a straight line is y = mx + b
m is the slope and b is the y-intercept.
the slope is -2, therefore the equation becomes y = -2x + b
solve for b by replacing x and y with the corresponding values from either one of the points.
i used (2,2).
y = -2x + b becomes 2 = -2 * 2 + b
solve for b to get b = 6
the equation of the line that represents the average rate of change between the two given points is g(x) = -2x + 6.
you can graph the equation of g(x) = x^2 - 2 and the equation of g(x) = -2x + 6
let y = g(x) for graphing purposes.
the graph looks like this.
the average rate of change is represented by the blue line.
the actual rate of change between each successive value of x is represented by the red line.