Linear vs. nonlinear functions Knowledge-Based Study

Study notes

1. What is a Linear Function?

   A function is classified as linear if its graph is a non-vertical straight line and its rate of change between any two points is strictly constant. In algebraic terms, a linear function can always be written in the form:

   $$f(x) = mx + b$$

   where $m$ represents the constant rate of change (the slope) and $b$ represents the initial value or $y$-intercept.

   What is a Nonlinear Function?

   A function is nonlinear if its rate of change varies across different intervals of its domain. The graph of a nonlinear function is curved rather than straight. Any function that cannot be rearranged into the form $f(x) = mx + b$ is nonlinear.

   Algebraic Characteristics

   To determine whether an equation defines a linear or nonlinear function, examine the exponents and operations applied to the independent variable $x$:

   • Linear equations: The variable $x$ has an exponent of exactly 1, does not appear in a denominator, is not under a radical sign, and is not an exponent itself. Examples include $y = -3x + 7$, $2x - 5y = 10$, and $f(x) = \frac{x}{4} + 1$.
   • Nonlinear equations: The variable $x$ appears with powers other than 1, inside absolute value bars, in a denominator, or as an exponent. Examples include:
     • Quadratic: $f(x) = 2x^2 - 4x + 1$
     • Exponential: $g(x) = 3^x$
     • Rational: $h(x) = \frac{5}{x}$
     • Radical: $k(x) = \sqrt{x + 2}$

   Using Tables of Values

   When given a table of values with equally spaced inputs ($x$-values), you can check whether a function is linear by calculating the change in output values ($\Delta y$).

   Linear Example

   Consider the table:

   • When $x = 0$, $y = 4$
   • When $x = 1$, $y = 7$ ($\Delta y = 3$)
   • When $x = 2$, $y = 10$ ($\Delta y = 3$)
   • When $x = 3$, $y = 13$ ($\Delta y = 3$)

   Because $\Delta x = 1$ is constant and $\Delta y = 3$ is constant, the rate of change $\frac{\Delta y}{\Delta x} = \frac{3}{1} = 3$ is uniform everywhere. This represents a linear function.

   Nonlinear Example

   Consider another table:

   • When $x = 1$, $y = 1$
   • When $x = 2$, $y = 4$ ($\Delta y = 3$)
   • When $x = 3$, $y = 9$ ($\Delta y = 5$)
   • When $x = 4$, $y = 16$ ($\Delta y = 7$)

   Although the inputs increase by a constant increment ($\Delta x = 1$), the differences between consecutive outputs are not equal (3, 5, 7). Because the rate of change is not constant, this represents a nonlinear function.

   Step-by-Step Decision Framework

   1. Graphical representation: Look at the graph. If it forms a single, continuous, straight line, it is linear. If it bends, curves, or breaks, it is nonlinear.
   2. Equation representation: Simplify the expression. If $x$ has an exponent of 1 and is not in an exponent, radical, or denominator, the function is linear.
   3. Tabular representation: Compute the ratio $\frac{\Delta y}{\Delta x}$ between several pairs of points. If this ratio is identical for all pairs, the function is linear; otherwise, it is nonlinear.