Linear vs. Quadratic Functions: Differentiation and Properties

Linear and quadratic functions are two fundamental types of mathematical functions that differ in their degree and properties. Here's a breakdown of the key differences between linear and quadratic functions:

1. Degree of the Function:

• Linear Function: A linear function has a degree of 1, which means it contains only variables raised to the first power. The general form of a linear function is $f(x) = ax + b$, where $a$ and $b$ are constants, and $x$ is the variable.
• Quadratic Function: A quadratic function has a degree of 2, which means it contains variables raised to the second power. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

2. Shape of the Graph:

• Linear Function: The graph of a linear function is a straight line. It has a constant slope, and the relationship between the dependent variable and the independent variable is always proportional.
• Quadratic Function: The graph of a quadratic function is a parabola. It has a curved shape, and the relationship between the dependent variable and the independent variable is nonlinear.

3. Rate of Change:

• Linear Function: In a linear function, the rate of change (slope) is constant. For every unit change in the independent variable, there is a constant change in the dependent variable.
• Quadratic Function: In a quadratic function, the rate of change is not constant. It varies with the value of the independent variable, leading to a curve in the graph.

4. Characteristics of the Graph:

• Linear Function: The graph of a linear function passes through the origin (0,0) if the constant term ($b$) is zero. Otherwise, it is a straight line with a slope determined by the coefficient $a$.
• Quadratic Function: The graph of a quadratic function does not necessarily pass through the origin. It has a vertex, which is the minimum or maximum point of the parabola, depending on the sign of the leading coefficient ($a$). The axis of symmetry is a vertical line that passes through the vertex.

5. Examples:

• Linear Function Example: $f(x) = 2x + 3$
  • This is a linear function with a slope of 2 and a y-intercept of 3. Its graph is a straight line.
• Quadratic Function Example: $f(x) = x^2 - 4x + 4$
  • This is a quadratic function with a vertex at (2,0). Its graph is a parabola opening upward.

In summary, the primary differences between linear and quadratic functions lie in their degree, graph shapes, rate of change, and characteristics. Linear functions produce straight-line graphs with constant slopes, while quadratic functions produce parabolic graphs with nonlinear relationships between variables.