Grades Grade 8 FunctionsLinear vs. Nonlinear Functions

Linear vs. Nonlinear Functions

🔗 Functions

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Linear vs. Nonlinear Functions

A function is linear if its rate of change is constant; nonlinear functions have changing rates of change.

A linear function has a constant rate of change (slope) and its graph is a straight line. A nonlinear function has a rate of change that varies — its graph curves. Common nonlinear examples include quadratic (y = x²) and exponential functions.

\text{Linear: } y = mx + b \qquad \text{Nonlinear: } y = x^2,\; y = 2^x,\; y = \frac{1}{x}

Linear Function

Constant slope (Δy/Δx is the same between any two points)Graph is a straight lineEquation: y = mx + bExample: distance = 60t (constant 60 mph)

Nonlinear Function

Slope changes between pointsGraph curves (parabola, exponential curve, hyperbola)Equation: y = x², y = 2ˣ, etc.Example: area of a square = s² (changes faster as s grows)

Test a table: if the differences in y are constant (constant Δy for constant Δx), the function is linear.

📋Identifying linear vs. nonlinear from a table

Table 1 — x: 0 1 2 3 | y: 1 3 5 7 → Δy = 2 each time → LINEAR
Table 2 — x: 0 1 2 3 | y: 1 1 4 9 → Δy = 0, 3, 5 (not constant) → NONLINEAR

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Remember This!

On a graph: a straight line = linear function. Any kind of curve (upward, downward, S-shaped) = nonlinear. This visual check is fast and reliable.

✏️ Try It!

A table shows x: 1, 2, 3, 4 and y: 2, 4, 8, 16. Is this function linear or nonlinear?

Practice with CalcVerse

📈Line Grapher (y = mx + b)📊Function Grapher

Take Quiz 📝 — 25 Questions