Linear and quadratic functions are both fundamental concepts in mathematics, but they differ significantly in their form, graph, and behavior.
Key Differences: Linear vs. Quadratic Functions
| Feature | Linear Function | Quadratic Function |
|---|---|---|
| General Form | y = mx + b | y = ax2 + bx + c |
| Graph | Straight line | Curved parabola |
| Slope | Constant | Varies depending on x |
| Degree | 1 (highest power of x is 1) | 2 (highest power of x is 2) |
| Y-intercept | 'b' in y = mx + b | 'c' in y = ax2 + bx + c |
| Roots/Zeros | At most one real root (x-intercept) | Can have zero, one, or two real roots (x-intercepts) |
Detailed Comparison
1. Form and Equation
- Linear Functions: These functions are represented by the equation y = mx + b, where 'm' is the slope (rate of change) and 'b' is the y-intercept (the point where the line crosses the y-axis). As the reference states, linear functions are typically in the form y = mx + b.
- Example: y = 2x + 3
- Quadratic Functions: These functions are represented by the equation y = ax2 + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The 'a' value determines if the parabola opens upward (a > 0) or downward (a < 0). According to the provided quadratic functions are typically in the form y = ax2 + bx + c.
- Example: y = x2 - 4x + 4
2. Graph
- Linear Functions: The graph of a linear function is always a straight line. To graph a linear function, you can start by plotting the y-intercept (b) and then use the slope (m) to find another point on the line.
- Quadratic Functions: The graph of a quadratic function is a parabola, which is a U-shaped curve. The vertex of the parabola is the point where the curve changes direction. The x-coordinate of the vertex can be found using the formula x = -b / 2a.
3. Rate of Change (Slope)
- Linear Functions: Linear functions have a constant rate of change, meaning the slope is the same everywhere along the line.
- Quadratic Functions: Quadratic functions have a rate of change that varies. The slope of the parabola changes as you move along the curve.
4. Roots (x-intercepts)
- Linear Functions: A linear function can have at most one root, which is the x-value where the line crosses the x-axis (where y = 0).
- Quadratic Functions: A quadratic function can have zero, one, or two real roots. These roots correspond to the x-values where the parabola intersects the x-axis.
5. Examples and Applications
-
Linear Functions:
- Modeling constant speed: The distance traveled by a car moving at a constant speed over time can be modeled using a linear function.
- Simple interest: The amount of money earned from simple interest over time is a linear function.
-
Quadratic Functions:
- Projectile motion: The height of a projectile (like a ball thrown in the air) over time can be modeled using a quadratic function.
- Area calculations: The area of a square as a function of its side length is a quadratic function (Area = side2).
6. Graphing
-
Linear Functions: Start at the y-intercept (b) and use the slope (m) to find the next point.
-
Quadratic Functions: Find the vertex using x = -b/2a. Then plug the x value back into the original equation to find the y value. Then pick 2 x values on either side of the vertex and calculate their corresponding y values. Finally, plot the points and create the parabola by connecting them.
