What's an example of a situation in which it could be modeled by a linear function, but not by a direct variation?
a car rental company charges $45 per day plus 15 cents per km.
cost = 0.15km + 45
To find an example of a situation that can be modeled by a linear function but not by a direct variation, it's helpful to understand the characteristics of these two types of functions.
A linear function is a mathematical equation in the form of y = mx + b, where m represents the slope of the line and b is the y-intercept. In a linear function, the relationship between the independent variable (x) and the dependent variable (y) is constant, and the line representing this relationship is always straight.
On the other hand, a direct variation is a special case of a linear function. It is a relationship in which the dependent variable is directly proportional to the independent variable. In other words, the equation is of the form y = kx, where k is a constant representing the rate of change.
Now, let's consider an example where a situation can be modeled by a linear function but not by a direct variation. Suppose you are tracking the distance traveled by a car over time. In the initial stages, the car is moving with a constant speed, but after a certain point, it starts accelerating.
In this scenario, the distance traveled (y) is determined by both the time elapsed (x) and the car's changing speed. As time increases, the speed of the car increases as well. Since the rate of change of the dependent variable (distance) cannot be expressed as a constant multiple of the independent variable (time), this situation cannot be modeled by a direct variation.
However, we can use a linear function to describe this situation by considering the changing rate of acceleration. We can define an equation such as y = mx + b, where m represents the changing rate of acceleration, and b is the initial distance when the car started moving.
Therefore, this is an example of a situation that can be modeled by a linear function but not by a direct variation, as the rate of change is not constant.