Question

Describing the Effect of the y-Intercept
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 1

Not necessarily. The y-intercept (value at time 0) only tells you who starts larger; what determines who is largest later is the rates of change (the slopes or how the functions change with time).

For linear functions f(t)=a1+m1t and g(t)=a2+m2t:
- If a1>a2 and m1≥m2, then f(t)>g(t) for all t≥0 (the one that starts larger and does not decrease faster stays larger).
- If a1>a2 but m1<m2, the faster-growing g will catch up. They meet when t = (a1−a2)/(m2−m1), and for larger t the one with larger slope is greater.

Example: f(t)=10−4t and g(t)=5−1t. At t=0 f(0)=10>5, but at t=2 f(2)=2 and g(2)=3, so g becomes larger after t= (10−5)/(−1−(−4)) = 5/3.

For non‑linear functions the same idea holds but functions can cross multiple times depending on shapes.

In the travelling-to-school context: someone who starts farther ahead (higher initial value) won’t necessarily stay ahead if another person travels faster (has a larger slope); conversely, if the faster person still has a lower rate, the starter will remain ahead.