Overview
The rate of change is a ratio that compares the change in values of the y variables to the change in values of the x variables. If the rate of change is constant and linear, the rate of change is the slope of the line. The slope of a line may be positive, negative, zero, or undefined.
Dependent and Independent Variables
Another way to look at the rate of change is to look at the definitions of the variables themselves. The y variables are the dependent variables and the x variables are the independent variables. Suppose that a worker is paid $10.00 per hour. The amount of the paycheck depends on the number of hours worked, so that worker is paid $160.00 for 16 hours one week, and $180.00 for 18 hours the next week. The change in the dependent variable is 180-160 or 20 and the change in the independent variable is 18-16, or 2, so the rate of change is 20/2, or 10/1.
Linear Rate of Change
If the relationship between the dependent and independent variables is linear, the rate of change is the same between any two sets of variables along the line. Suppose that the worker in the above example is paid $50.00 for 5 hours one week and $80.00 for 8 hours the next week. The change in the dependent variable is 80-50, or 30, and the change in the independent variable is 8-5, or 3. The rate of change is 30/3, or 10/1, the same rate of change as in the first example. In a linear relationship, the rate of change is constant.
Type of Slope
The slope of a line can also be described by its direction. In the relationship of the paycheck and number of hours worked, the line rises from left to right. The more hours worked, the higher the paycheck, with a positive slope. Suppose a plane is landing. The relationship between its elevation and the time from its highest altitude is a falling line from left to right, a negative slope. If the relationship is a horizontal line, so that no change occurs, the slope is zero. However, if the relationship is a vertical line, the slope is undefined. It is also not a function, as there are multiple values of y for one value of x.
Slope Formula
The slope of a line is usually represented by the variable m. It is expressed by the ratio of the difference in value of y variables to the difference in value of x variables. In algebraic terms, if (x1, y1) (x2, y2) are the coordinates of any two points on a line, then m= (y2 – y1)/(x2 – x1).
Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Grand Forks, ND: visit: Tutoring in Grand Forks, ND