When comparing functions, such as Seth’s scenarios for travelling to school, will the function with the greatest initial value continue to be the greatest as time increases? Explain why.

Answer:

No, the function with the greatest initial value will not necessarily remain the greatest as time increases. Whether a function stays "on top" depends on its rate of change (slope) relative to the other functions. Explanation :Initial Value (\(y\)-intercept): This only tells you where the function starts at \(t = 0\).Rate of Change (Slope): This determines how fast the value grows or shrinks over time .Scenarios :Lower Rate of Change: If the function with the higher starting point has a smaller slope (or a negative slope) than another function, the other function will eventually "catch up" and overtake it. Different Function Types: In travel scenarios, one person might start further ahead (higher initial value) but walk slowly, while another person starts at home (zero initial value) but bikes quickly. The biker will eventually pass the walker.