Rate of change of linear function is constant whereas the rate of change of the quadratic functions are not constant. The rate of change is obtained by differentiating the linear as well as quadratic functions.
In this article we will explore the rate of change in linear and quadratic functions in detail.
Table of Content
What is Linear Function?
A linear function is a function that creates a straight line when graphed. It has the general form:
f(x) = mx + b
where:
- m is the slope of the line and
- b is the y-intercept the point where the line crosses y-axis.
The slope m represents a constant rate of change meaning that for the each unit increase in x. f(x) changes by the constant value. Linear functions have a steady unchanging rate of the growth or decay, and their graph is always a straight line.
What is Quadratic Function?
A quadratic function is a type of the polynomial function with the general form:
f(x) = ax2+ bx + c
where a, b and c are constants and 𝑎≠0. The graph of the quadratic function is a parabola in which can open upwards or downwards depending on the sign of a. The rate of change in the quadratic functions is not constant it increases or decreases depending on the value of the x.
Rate of Change in Linear and Quadratic Functions
The rate of change is determined by finding the derivative of the given function. In this section we will learn about the rate of change in linear and quadratic functions.
Rates of Change in Linear Functions
In linear functions, the rate of change is constant. A linear function can be expressed in the form f(x) = mx + b where m is the slope of the line, and b is the y-intercept. The slope m represents the rate of change.
Rate of Change in Linear Functions Formula
The rate of change for the linear function is calculated as:
Rate of Change= Δx\Δy
where
- Δy is the change in the y-values and
- Δx is the change in the x-values.
Rates of Change in Quadratic Functions
Quadratic functions exhibit a variable rate of the change unlike the linear functions. A quadratic function is given by f(x) = ax2 + bx + c. The rate of change at any point is not constant and varies with the x.
Rate of Change in Quadratic Functions Formula
To find the rate of change for the quadratic function we use the derivative. The derivative of the f(x) = ax2 + bx + c is:
f(x) = 2ax + b
This derivative represents the instantaneous rate of the change at any point x.
Solved Examples on Rates of Change in Linear and Quadratic Functions
Example 1: Given the linear function f(x) = 4x - 3 find the rate of change.
Solution:
For a linear function f(x) = mx + b the rate of change is represented by the slope m.
Here,
- f(x) = 4x - 3 so the slope m = 4.
Thus, the rate of change is 4. This means for each unit increase in x, f(x) increases by 4 units.
Example 2: Interpret the rate of change of the linear function f(x) = -2x + 7.
Solution:
The linear function f(x) = -2x + 7 has a slope m = -2.
The rate of change is -2 indicating that for each unit increase in x, f(x) decreases by the 2 units.
Example 3: For the quadratic function f(x) = 3x2 - 5x + 2 calculate the rate of the change at x = 4.
Solution:
First, find the derivative
f'(x) = 6x - 5
At x = 4:
f'(4) = 6⋅4 -5
⇒ f'(4) = 24 - 5
⇒ f'(4) = 19
Thus, the rate of change at x=4 is 19.
Example 4: A ball is thrown upwards and its height in the meters is given by the h(t) = - 5t2 + 20t + 2 where t is the time in seconds. Find the rate of change of the height at t = 2 seconds.
Solution:
First, find the derivative
h'(t) = -10t + 20
At t = 2:
h'(2) = -10⋅2 + 20
⇒ h'(2) = -20 + 20
⇒ h'(2) = 0
The rate of change of the height at t=2 seconds is 0 meters per second meaning the ball is at its maximum height at this time.
Example 5: A company’s revenue function is given by R(x) = 100x - 0.5x2 where x is the number of the units sold. Find the rate of change of the revenue when x = 10 units.
Solution:
First, find the derivative
R'(x) = 100 - x
At x = 10:
R'(10) = 100 - 10
⇒ R'(10) = 90
Thus, the rate of change of the revenue when 10 units are sold is 90 dollars per unit.
Practice Questions on Rates of Change
Q1. For the linear function f(x) = 7x - 4 determine the rate of change.
Q2. Find the rate of change of the quadratic function g(x) = 2x2 + 3x - 1 at x = 5.
Q3. A car’s speed is modeled by the function s(t) = 2t2 + 3t, where t is time in hours. What is the rate of change of the speed at t = 3 hours?
Q4. For the quadratic function f(x) = -x2 + 4x + 1 find the rate of change at x = 2.
Q5. Determine the rate of change of the linear function f(x) = -5x + 8.
Q6. A rocket’s height is given by h(t) = 15t - 4.5t2 where t is time in seconds. Find the rate of change of the height at t = 4 seconds.
Q7. Given the function f(x) = 6x - 2x2 find the rate of the change at x = 1.
Q8. For the function f(x) = 5x2 - 3x + 7 calculate the rate of the change at x = 2.
Q9. A firm’s profit function is P(x) = 50x -x2. What is the rate of change in profit when x = 6 units are produced?
Q10. Determine the rate of change of the quadratic function f(x) = x2 -6x + 9 at x = 3.
Answer Key
- 7
- 23
- 15
- 0
- -5
- -21
- 2
- 17
- 38
- 0
