Analyzing Data for Optimal Function Modeling
The image contains a question asking which type of function would best model the given data, providing options A. Linear, B. Quadratic, C. Exponential, and D. Absolute value. The data provided in the table shows the population of a city in thousands (y) for time in years (x) after the beginning of a decade.
The data is as follows:
- Time (years), x: 0, 1, 2, 6, 8
- Population (thousands), y: 52, 57, 137, 152, 227
To determine the best model for this data, we must examine how the population changes as time increases. A linear function is represented by a straight line and involves a constant rate of change. A quadratic function describes a parabolic relationship and involves acceleration or deceleration of the rate of change. An exponential function involves growth or decay that increases or decreases at a nonlinear rate, typically becoming more pronounced as time goes on. An absolute value function would typically create a V-shaped graph, which does not seem to be indicative of the data points.
Looking at the intervals:
- From x=0 to x=1, the population grows from 52 to 57 (a difference of 5).
- From x=1 to x=2, the population grows from 57 to 137 (a difference of 80).
- From x=2 to x=6, the population grows from 137 to 152 (a difference of 15 over 4 years).
- From x=6 to x=8, the population grows from 152 to 227 (a difference of 75 over 2 years).
Considering the significant jump from 57 to 137 and then the more gradual changes followed by another large jump, this suggests that an exponential growth trend might be more suitable. A linear or quadratic model would not typically show the sharp increases seen between certain intervals (like from year 1 to year 2), and absolute value is not reflective of the increasing growth pattern.
Therefore, the type of function that would best model this data is C. Exponential.
