Eighth Grade Mathematics: Distinguishing Linear from Nonlinear Functions
In mathematics, functions are broadly categorized into two fundamental kingdoms: linear functions and nonlinear functions. A linear function describes an unvarying, constant rate of change that produces a perfectly straight line on a graph, while a nonlinear function describes a rate of change that varies, bends, curves, or accelerates. In this chapter, you will learn to distinguish linear from nonlinear functions across algebraic equations, data tables, and coordinate graphs, and discover why the constant rate of change is the hallmark of linearity.
The Hallmark of Linearity: Constant Rate of Change
The defining mathematical characteristic of a linear function is that its rate of change (the ratio of the change in y to the change in x) is strictly constant across its entire domain.
Linear Criterion:
Delta y
------- = m (A single unchanging constant value for ALL intervals!)
Delta x
If the rate of change between any two points differs even slightly from the rate of change between another pair of points, the function is nonlinear.
Table Comparison:
Linear Function Table: Nonlinear Function Table:
x y Delta y / Delta x x y Delta y / Delta x
0 2 - 0 0 -
1 5 (5-2)/(1-0) = 3 1 1 (1-0)/(1-0) = 1
2 8 (8-5)/(2-1) = 3 2 4 (4-1)/(2-1) = 3 (Different!)
3 11 (11-8)/(3-2) = 3 3 9 (9-4)/(3-2) = 5 (Different!)
Constant rate of change = 3. Rate of change increases: 1, 3, 5.
This table is LINEAR. This table is NONLINEAR (Quadratic).
In the linear table, y increases by 3 every time x increases by 1. In the nonlinear table, y increases by 1, then by 3, then by 5; because the rate of change is changing, the function cannot be linear.
Identifying Linearity from Equations
The algebraic equation of a linear function can always be written in the standard slope-intercept form:
Linear Form: y = mx + b
In a linear equation, the variable x must be raised strictly to the first power (x^1). There can be no exponents other than 1 on the variable, no variables multiplied together, no variables in the denominator of a fraction, and no variables under radical signs.
Linear Equations: Nonlinear Equations:
y = 4x - 7 (m = 4, b = -7) y = x^2 + 5 (x is squared)
y = -0.5x (m = -0.5, b = 0) y = 2^x (x is in the exponent)
y = (2/3)x + 1 (m = 2/3, b = 1) y = 6 / x (x is in denominator)
2x + 3y = 12 (Standard form) y = sqrt(x) (x is under radical)
y = 8 (Horizontal line) y = x^3 - 2x (x is cubed)
If an equation contains terms like x^2, 3^x, 1/x, or sqrt(x), its graph will curve or bend, making it nonlinear.
Graphical Distinction: Straight Lines versus Curves
Graphically, the difference between linear and nonlinear functions is immediately apparent:
Linear Graph (Straight): Nonlinear Graph (Curved):
y y
8 | / 8 | \ / (Parabola)
6 | / 6 | \ /
4 | / 4 | \ /
2 | / 2 | \_/
0 +-------------> x 0 +-------------> x
0 1 2 3 4 -2 -1 0 1 2
A linear graph is a single continuous straight line without bends, vertices, breaks, or curvature. Any graph that curves, forms a U-shape (parabola), forms an S-curve (cubic), or forms a V-shape (absolute value) represents a nonlinear relation.
Chapter Exercises
Exercise 1: Determine whether the function represented by the table is linear or nonlinear: for x = 1, y = 12; for x = 3, y = 20; for x = 6, y = 32; and for x = 10, y = 48. Calculate the rate of change Delta y / Delta x across each interval to justify your answer.
Exercise 2: Classify each of the following equations as linear or nonlinear, providing a specific algebraic reason for each: y = 5 - 2x, y = 4/(x + 1), y = 3x^2 - 7, and x/4 + y/3 = 1.
Exercise 3: Explain in paragraph form why the area of a square as a function of its side length (A = s^2) is nonlinear, while the perimeter of a square as a function of its side length (P = 4s) is linear, discussing both rates of change and graphical shapes.
Solutions and Explanations
Solution 1: To determine whether the table represents a linear or nonlinear function, calculate the rate of change Delta y / Delta x between consecutive pairs of coordinates. Between (1, 12) and (3, 20): Delta y / Delta x = (20 - 12) / (3 - 1) = 8 / 2 = 4. Between (3, 20) and (6, 32): Delta y / Delta x = (32 - 20) / (6 - 3) = 12 / 3 = 4. Between (6, 32) and (10, 48): Delta y / Delta x = (48 - 32) / (10 - 6) = 16 / 4 = 4. Because the rate of change evaluates to exactly 4 across every interval regardless of the spacing between x-values, the function possesses a constant rate of change. Therefore, the function is strictly linear, with equation y = 4x + 8.
Solution 2: The equation y = 5 - 2x is linear because it can be rewritten in slope-intercept form as y = -2x + 5, where x is raised to the first power with a constant slope of -2. The equation y = 4/(x + 1) is nonlinear because the variable x resides in the denominator of a fraction, which creates a rational curve with a varying rate of change. The equation y = 3x^2 - 7 is nonlinear because the variable x is raised to the second power (exponent of 2), which produces a curved U-shaped parabola. The equation x/4 + y/3 = 1 is linear because it can be rewritten in slope-intercept form: multiplying by 12 gives 3x + 4y = 12, and isolating y yields y = (-3/4)x + 3, where x is raised to the first power with a constant slope of -3/4.
Solution 3: The perimeter function P = 4s is linear because the side length s is raised to the first power and multiplied by the constant 4. The rate of change of perimeter relative to side length is Delta P / Delta s = 4; for every 1-unit increase in side length, the perimeter always increases by exactly 4 units, regardless of whether the square is small or large. Graphed on coordinate axes, P = 4s forms a perfectly straight ray originating from (0, 0). In contrast, the area function A = s^2 is nonlinear because the variable s is squared. The rate of change of area is not constant: when side length increases from 1 to 2, area increases by 3 (from 1 to 4); when side length increases from 2 to 3, area increases by 5 (from 4 to 9); and when side length increases from 3 to 4, area increases by 7 (from 9 to 16). Because the rate of change increases continuously as the side length grows, the graph of A = s^2 forms an accelerating upward curve rather than a straight line.