Final answer:
Linear functions, represented by the equation y = mx + b, model the relationship between two variables and can be used to make predictions and solve problems by determining how one variable affects another.
Step-by-step explanation:
Linear functions can be powerful tools in modeling situations and solving a wide range of problems. These functions are typically represented by the algebraic equation y = mx + b, where 'm' represents the slope and 'b' indicates the y-intercept. The slope describes the rate of change between the variables, indicating how much y will change as x increases by one unit. The y-intercept is the value of y when x is zero, essentially where the line crosses the y-axis on a graph.
For instance, in modeling the relationship between two variables, such as the number of flu cases over a period of years, we can set the year as the independent variable x and the number of flu cases as the dependent variable y. By finding a linear pattern in historical data, we can establish a linear equation that describes this relationship, enabling predictions for future years.
As another example, consider the equation y = 3,000x + 500 which models the total cost of attending a college (y) based on the number of years enrolled (x). Here, the slope 3,000 indicates the additional cost per year, and 500 could be a fixed cost that occurs regardless of the number of years enrolled.