Linear vs. Quadratic

Attribute	Linear	Quadratic
Equation Form	y = mx + b	y = ax^2 + bx + c
Graph Shape	Straight line	Parabola
Number of Solutions	One	Two
Vertex	N/A	(h, k)
Axis of Symmetry	N/A	x = h

Introduction

Linear and quadratic functions are two of the most common types of functions studied in mathematics. While both are used to model relationships between variables, they have distinct attributes that set them apart. In this article, we will explore the differences between linear and quadratic functions in terms of their equations, graphs, and applications.

Equations

A linear function is represented by an equation of the form y = mx + b, where m is the slope of the line and b is the y-intercept. The graph of a linear function is a straight line that extends infinitely in both directions. On the other hand, a quadratic function is represented by an equation of the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola that opens either upwards or downwards.

Graphs

When comparing the graphs of linear and quadratic functions, one can observe distinct differences. The graph of a linear function is a straight line with a constant slope, while the graph of a quadratic function is a curved parabola. Linear functions have a constant rate of change, resulting in a straight line, whereas quadratic functions have a variable rate of change, leading to the curved shape of a parabola.

Applications

Linear functions are commonly used to model relationships that have a constant rate of change, such as distance vs. time or cost vs. quantity. For example, the equation y = 2x + 5 represents a linear function where y is the total cost, x is the quantity purchased, and 2 is the cost per unit. On the other hand, quadratic functions are used to model relationships that have a non-linear rate of change, such as projectile motion or the trajectory of a thrown object. The equation y = x^2 - 3x + 2 represents a quadratic function where y is the height of the object and x is the time elapsed.

Key Differences

One key difference between linear and quadratic functions is the degree of the highest power of the variable in the equation. Linear functions have a degree of 1, while quadratic functions have a degree of 2. This difference in degree results in the distinct shapes of their graphs and the nature of their relationships. Additionally, linear functions have a constant rate of change, while quadratic functions have a variable rate of change.

Conclusion

In conclusion, linear and quadratic functions have unique attributes that distinguish them from each other. While linear functions are characterized by straight lines and constant rates of change, quadratic functions exhibit curved parabolas and variable rates of change. Understanding the differences between these two types of functions is essential for solving mathematical problems and analyzing real-world relationships.