Question
A regional train passes by a certain train station halfway along its trip each day. The graph models the train traveling at a constant speed. Which equation best represents the graph?
f(x) = |x| + 100
f(x) = |100x|
f(x) = |x + 100|
f(x) = |100 − x|
f(x) = |x| + 100
f(x) = |100x|
f(x) = |x + 100|
f(x) = |100 − x|
Answers
GPT-4o mini
To determine which equation best represents the graph of a regional train traveling at a constant speed, we should analyze the characteristics of the equations provided.
1. **f(x) = |x| + 100**: This function is a V-shaped graph that opens upwards, with its vertex at (0, 100). It does not represent constant speed as it increases at an increasing rate.
2. **f(x) = |100x|**: This function also creates a V-shape that opens upwards, but it scales with the variable x. The graph becomes steeper as x increases, which again does not represent constant speed.
3. **f(x) = |x + 100|**: This function is also V-shaped with its vertex at (-100, 0). It still doesn't represent constant speed as the slope changes depending on the value of x.
4. **f(x) = |100 - x|**: This equation creates a V-shape that opens upwards with a vertex at (100, 0). Up to x = 100, the value of f(x) decreases linearly, and beyond x = 100, the value increases linearly, which could represent the journey of a train going towards a certain point and then returning.
Given that the question states the train is traveling at a constant speed and passes a certain point midway, the fourth equation, **f(x) = |100 - x|**, is the most reasonable representation. It models the distance at a constant rate until it reaches the midpoint, then returns.
So, the best equation representing the graph for the train traveling at a constant speed is:
**f(x) = |100 - x|**.
1. **f(x) = |x| + 100**: This function is a V-shaped graph that opens upwards, with its vertex at (0, 100). It does not represent constant speed as it increases at an increasing rate.
2. **f(x) = |100x|**: This function also creates a V-shape that opens upwards, but it scales with the variable x. The graph becomes steeper as x increases, which again does not represent constant speed.
3. **f(x) = |x + 100|**: This function is also V-shaped with its vertex at (-100, 0). It still doesn't represent constant speed as the slope changes depending on the value of x.
4. **f(x) = |100 - x|**: This equation creates a V-shape that opens upwards with a vertex at (100, 0). Up to x = 100, the value of f(x) decreases linearly, and beyond x = 100, the value increases linearly, which could represent the journey of a train going towards a certain point and then returning.
Given that the question states the train is traveling at a constant speed and passes a certain point midway, the fourth equation, **f(x) = |100 - x|**, is the most reasonable representation. It models the distance at a constant rate until it reaches the midpoint, then returns.
So, the best equation representing the graph for the train traveling at a constant speed is:
**f(x) = |100 - x|**.