1st Degree Function Change Rate

In a 1st degree function we have that the rate of change is given by the coefficient a. We have that a 1st degree function respects the following formation law f (x) = ax + b, where a and b are real numbers and b ≠ 0. The rate of change of the function is given by the following expression:


Example 1

Let's go through a demonstration to prove that the rate of change of the function f(x) = 2x + 3 is given by 2.
f (x) = 2x + 3
f (x + h) = 2 * (x + h) + 3 → f (x + h) = 2x + 2h + 3 (h ≠ 0)
So we have to:
f (x + h) − f (x) = 2x + 2h + 3 – (2x + 3)
f (x + h) − f (x) = 2x + 2h + 3 – 2x – 3
f (x + h) − f (x) = 2h
Then:

Note that after the demonstration we find that the rate of change can be calculated directly by identifying the value of the coefficient a in the given function. For example, in the following functions the rate of change is given by:
a) f (x) = –5x + 10, rate of change a = –5
b) f (x) = 10x + 52, rate of change a = 10
c) f (x) = 0.2x + 0.03, rate of change a = 0.2
d) f (x) = –15x – 12, rate of change a = –15


Example 2

See one more demonstration proving that the rate of change of a function is given by the slope of the line. The given function is as follows: f (x) = –0.3x + 6.
f (x) = -0.3x + 6
f (x + h) = –0.3(x + h) + 6 → f (x + h) = –0.3x –0.3h + 6
f (x + h) − f (x) = –0.3x –0.3h + 6 – (–0.3x + 6)
f (x + h) − f (x) = –0.3x –0.3h + 6 + 0.3x – 6
f (x + h) − f (x) = –0.3h

Do not stop now... There's more after the advertising ;)

The rate of change of a 1st degree function is determined in higher education courses by developing the derivative of a function. For such application we need to study some fundamentals involving notions of Calculus I. But let's demonstrate a simpler situation involving the derivative of a function. For this, consider the following statements:
The derivative of a constant value is equal to zero. For example:

f (x) = 2 → f’(x) = 0 (read f line)
The derivative of a power is given by the expression:

f(x) = x² → f’(x) = 2*x2–1 → f’(x) = 2x
f (x) = 2x³ – 2 → f’(x) = 3*2x3–1 → f’(x) = 6x²
Therefore, to determine the derivative (rate of change) of a 1st degree function, we just apply the two definitions shown above. Watch:
f (x) = 2x – 6 → f’(x) = 1*2x1–1 → f’(x) = 2x0 → f’(x) = 2
f (x) = –3x + 7 → f’(x) = –3

by Mark Noah
Graduated in Mathematics
Brazil School Team

1st Degree Function - Math - Brazil School

Would you like to reference this text in a school or academic work? Look:

SILVA, Marcos Noé Pedro da. "Rate of Variation of 1st Degree Function"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/taxa-variacao-funcao-1-o-grau.htm. Accessed on June 29, 2021.

Polynomial function: what is it, examples, graphs

Polynomial function: what is it, examples, graphs

A function is called polynomial function when its formation law is a polynomial. Polynomial funct...

read more
Exponential function: types, graph, exercises

Exponential function: types, graph, exercises

THE exponential function occurs when, in its formation law, the variable is in the exponent, with...

read more
Relation of the parabola to the delta of the second degree function

Relation of the parabola to the delta of the second degree function

The parabola is the graph of the function of the second degree (f (x) = ax2 + bx + c), also calle...

read more