The concept of graphing equations is an integral part of mathematics, and it is often important to determine which equation has the steepest graph. Generally, this is done by comparing the slopes of different equations, which can be calculated by taking the derivative of the equation. Depending on the equation, the steepest graph can vary, and it is important to understand the different types of equations and their implications.
Linear Equations
Linear equations are equations whose graphs are straight lines. Examples of linear equations are y = x and y = 2x, where y is the dependent variable and x is the independent variable. The slope of a linear equation is constant and can be calculated by taking the derivative of the equation. For example, the slope of the equation y = x is 1, and the slope of the equation y = 2x is 2. Therefore, the equation y = 2x has the steepest graph among the two.Quadratic Equations
Quadratic equations are equations whose graphs are parabolas. Examples of quadratic equations are y = x2 and y = 4x2, where y is the dependent variable and x is the independent variable. The slope of a quadratic equation is not constant and can be calculated by taking the derivative of the equation. For example, the slope of the equation y = x2 is 2x, and the slope of the equation y = 4x2 is 8x. Therefore, the equation y = 4x2 has the steepest graph among the two.Comparison
| Equation | Slope | |
|---|---|---|
| Linear | y = x | 1 |
| Linear | y = 2x | 2 |
| Quadratic | y = x2 | 2x |
| Quadratic | y = 4x2 | 8x |