Lesson 4: Comparing Linear and Nonlinear Functions

Property

To determine linearity from a table, you must calculate the rate of change ($\frac{\Delta y}{\Delta x}$) between consecutive points.

• If this ratio simplifies to the exact same number everywhere, the function is linear.
• If the ratio changes, the function is nonlinear.

Examples

• Linear Table: x values are (0, 1, 2, 3), y values are (2, 5, 8, 11). The rate of change is 3/1 = 3 between every single point.
• Nonlinear Table: x values are (0, 1, 2, 3), y values are (0, 1, 4, 9). The rate of change goes from 1/1 to 3/1 to 5/1. Because the rate keeps changing, it is a curve.

Explanation

When checking a table, a common trap is only looking at how much the y-values jump. You must always divide the jump in 'y' by the jump in 'x' for every single step. If that final fraction stays exactly the same, you have a straight line!

Property

Not all equations lead to a graph of values lying on a straight line. Some relationships, such as those where variables are multiplied to equal a constant (like $xy=k$) or added to equal a constant ($x+y=k$), produce graphs that are curves or lines with different characteristics than those from simple ratio or additive relationships.

Examples

• Two numbers, $x$ and $y$, must add up to 12 ($x+y=12$). Possible pairs are (2, 10), (5, 7), and (9, 3). The graph of these points is a line that slopes downward.
• A rectangle must have an area of 36 square inches ($L \times W = 36$). Possible pairs for (Length, Width) are (3, 12), (6, 6), and (9, 4). The graph of these points forms a curve.
• For the equation $y = x^2$, the relationship is not linear. Some pairs are (2, 4), (3, 9), and (5, 25). The graph is a curve that gets progressively steeper.

Explanation

Be aware that not all two-variable equations create straight-line graphs. Relationships like a fixed area ($L \times W = 24$) or a fixed sum ($x+y=10$) can create curves or lines that slope downwards, revealing different kinds of patterns.

Property

A function is linear if its equation can be written in the form $y = mx + b$. If an equation cannot be written in this form, it represents a nonlinear function.

Examples