# Linear Functions and Growth Rate

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Let [math]f(x) = mx+b[/math] be a real-valued linear function. The following questions will investigate the growth of this linear function, or the value [math]Delta f = f(x + Delta x) - f(x)[/math], over equal intervals of length [math]Delta x[/math], where [math]Delta x > 0[/math].

A.

What is the value of [math]Delta f = f(x + Delta x) - f(x)[/math], using the definition of [math]f(x) ?[/math] Simplify your answer fully.

- [math]m Delta x + 2b[/math]
- [math]Delta x - 2b[/math]
- [math]m Delta x[/math]
- [math]Delta x + 2m[/math]

B.

Given the answer in the previous question, which of the following gives the best reasoning as to why one can conclude that a linear function grows by equal differences over equal intervals?

- Because the resulting equation for [math]Delta f[/math] is also linear, it will increase at a constant rate.
- Because there is no slope in the resulting equation, the value of [math]Delta f[/math] is constant.
- Since the resulting equation for [math]Delta f[/math] has no [math]b[/math] value, it is independent of the y-intercept. As such, [math]Delta f[/math] will increase by equal amounts over equal intervals.
- Since [math]Delta f[/math] is dependent only on the length of the interval, as long as the interval [math]Delta x[/math] is constant, [math]Delta f[/math] will be the same.

C.

What if, for the linear function [math]f(x)[/math], [math]m =0 ?[/math]

- This means that the above reasoning is invalid.
- For how the function is defined above, [math]m[/math] cannot equal zero.
- It makes things easier, since the growth rate of [math]f(x)[/math] simply becomes zero (and thus constant) for all intervals.
- It does not change anything, since [math]Delta f[/math] is not dependent on [math]m[/math].