The document shows that if a function f has a constant average rate of change c between any two points, then f is a linear function. It proves that the equation f(x) = cx + (f(a) - ca) holds, where c is the slope of the linear function and f(a) - ca is the y-intercept. This establishes f(x) is linear with slope c and y-intercept f(a) - ca.

1. Suppose that the function f has the same average rate of change c between any two points, Find
the average rate of change of f between the points a and x to show that c= f(x) - f(a)/x - a. The
rate of change between any two points is Rearrange the equation in part (a) to show that f(x) =
cx + (f(a) - ca). Multiplying the equation in part(a) by x - a, we obtain f(x) - f(a) = c
Rearranging and adding f(a) to both sides, we have f(x) = cx + (f(a) - ,as desired. How does this
show that f is a linear function? What is the slope, and what is the y-intercept? Because this
equation is of form f(x) = Ax + B with constants and B = f(a) -, it represents a linear function
with slope and y-intercept f(a)s
Solution
a) Average rate of change between any two points x1 and x2 is given by:
Avg rate of change = {f(x2) -f(x1)} /(x2-x1)
Let x1 =a and x2 =x:
Avg rate of change = ={ f(x) - f(c)/( x-a) = c
c ={ f(x) - f(c)/( x-a)
b) Multiply aboveequation by ( x-a):
c(x-a) ={ f(x) - f(a)/( x-a)}(x-a)
cx - ca = f(x) - f(c)
f(x) - f(a) = c( x-a)
f(x) = f(a) + cx -ca
f(x) = cx + ( f(a) - ca )
Std linear function : y = mx +c where m is slope and c is a constant term
So, comparing with this equation we have :: f(x) = cx + ( f(a) - ca )
Its a linear equation
B = f(a) - ca
slope = c ad y-intercept = f(a) -ca
where m = c (slope)