Explanation:
Linear, exponential, and quadratic functions have different growth rates, which can be compared using the following methods:
Slope: The slope of a linear function represents its growth rate. If the slope is positive, the function is increasing at a constant rate. If the slope is negative, the function is decreasing at a constant rate. A steeper slope indicates a faster growth rate.
Growth factor: An exponential function has a constant growth factor, which represents the rate at which the function grows. For example, if an exponential function has a growth factor of 1.1, it will increase by 10% each time the independent variable increases by 1. A larger growth factor indicates a faster growth rate.
Concavity: A quadratic function has a constant rate of change that increases at a constant rate. The concavity of the parabola indicates the rate at which the function grows. If the concavity is positive, the function is increasing at an increasing rate (i.e., accelerating growth). If the concavity is negative, the function is increasing at a decreasing rate (i.e., decelerating growth).
Intercepts: The x-intercepts of a linear function, the y-intercepts of an exponential function, and the x-intercepts of a quadratic function can provide insight into their growth rates. For example, a linear function with a large positive y-intercept indicates that it is already quite large at the start, while a quadratic function with a negative x-intercept indicates that it starts decreasing after a certain point.
By considering these factors, we can compare the growth rates of linear, exponential, and quadratic functions and gain insight into their behavior over time.