Question
Make a graph on a sheet of paper. Then answer the question.The function d=|400−40t| can be used to find the distance a car is from you when starting 400 feet away and traveling 40 feet per second after t seconds.What do the key features of the function reveal?(1 point)ResponsesThe function is an increasing function, so as the time increases, the distance the car is from you decreases at a constant rate.The function is an increasing function, so as the time increases, the distance the car is from you decreases at a constant rate.The range is d≤0, so the distance the car is from you can only be negative values.The range is d is less than or equal to 0 , so the distance the car is from you can only be negative values.The domain is t≤0, so the time the car is traveling can only be negative values.The domain is t is less than or equal to 0 , so the time the car is traveling can only be negative values.There is a vertex at (10,0) which indicates that after 10 seconds the car is at the same location as you.There is a vertex at open paren 10 comma 0 close paren which indicates that after 10 seconds the car is at the same location as you.Graphing Calculator
Answers
GPT-4o mini
To analyze the function \( d = |400 - 40t| \), let’s break it down:
1. **Understanding the Function**:
- The function represents the absolute distance of a car from you as it travels towards you from an initial distance of 400 feet at a speed of 40 feet per second.
- The expression \( 400 - 40t \) gives the distance as a linear function, and the absolute value ensures that the distance is non-negative.
2. **Finding Key Features**:
- **Vertex**:
- The vertex occurs where the inside of the absolute value equals zero:
\[
400 - 40t = 0 \implies 40t = 400 \implies t = 10 \text{ seconds}
\]
- At this point, \( d = 0 \). Thus, the vertex is at \( (10, 0) \). This means that after 10 seconds, the car has reached your location.
- **Behavior of the Function**:
- For \( t < 10 \): The term \( 400 - 40t \) is positive, so \( d = 400 - 40t \), which means the distance is decreasing as time increases.
- For \( t > 10 \): The term \( 400 - 40t \) becomes negative, so \( d = 40t - 400 \), which means the distance is increasing as the car moves away from you after passing your position.
3. **Domain and Range**:
- **Domain**: Since \( t \) represents time, it can take any non-negative value: \( t \geq 0 \).
- **Range**: Since distance cannot be negative, the range is \( d \geq 0 \).
4. **Conclusion**:
- Based on these interpretations, the key features of the function reveal that the car approaches you for the first 10 seconds, reaches you at \( t = 10 \) seconds, and then starts moving away.
-
Given your original options:
- The correct response is:
- **"There is a vertex at \( (10,0) \), which indicates that after 10 seconds the car is at the same location as you."**
### Summary
1. **Vertex**: \( (10, 0) \)
2. **Domain**: \( t \geq 0 \)
3. **Range**: \( d \geq 0 \)
4. **Behavior**: Distance decreases until \( t = 10 \), then increases.
1. **Understanding the Function**:
- The function represents the absolute distance of a car from you as it travels towards you from an initial distance of 400 feet at a speed of 40 feet per second.
- The expression \( 400 - 40t \) gives the distance as a linear function, and the absolute value ensures that the distance is non-negative.
2. **Finding Key Features**:
- **Vertex**:
- The vertex occurs where the inside of the absolute value equals zero:
\[
400 - 40t = 0 \implies 40t = 400 \implies t = 10 \text{ seconds}
\]
- At this point, \( d = 0 \). Thus, the vertex is at \( (10, 0) \). This means that after 10 seconds, the car has reached your location.
- **Behavior of the Function**:
- For \( t < 10 \): The term \( 400 - 40t \) is positive, so \( d = 400 - 40t \), which means the distance is decreasing as time increases.
- For \( t > 10 \): The term \( 400 - 40t \) becomes negative, so \( d = 40t - 400 \), which means the distance is increasing as the car moves away from you after passing your position.
3. **Domain and Range**:
- **Domain**: Since \( t \) represents time, it can take any non-negative value: \( t \geq 0 \).
- **Range**: Since distance cannot be negative, the range is \( d \geq 0 \).
4. **Conclusion**:
- Based on these interpretations, the key features of the function reveal that the car approaches you for the first 10 seconds, reaches you at \( t = 10 \) seconds, and then starts moving away.
-
Given your original options:
- The correct response is:
- **"There is a vertex at \( (10,0) \), which indicates that after 10 seconds the car is at the same location as you."**
### Summary
1. **Vertex**: \( (10, 0) \)
2. **Domain**: \( t \geq 0 \)
3. **Range**: \( d \geq 0 \)
4. **Behavior**: Distance decreases until \( t = 10 \), then increases.