Make a graph on a sheet of paper. Then answer the question.

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)
Responses

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 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 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.

Answer 1

To analyze the function \( d = |400 - 40t| \), let's start by understanding its key features:

1. Distance Interpretation: The equation models the distance \( d \) from you at the start (when \( t = 0 \)) with the car initially 400 feet away from you, traveling towards you at a speed of 40 feet per second.

2. Finding the Vertex: The expression inside the absolute value, \( 400 - 40t \), becomes zero when you set it equal to zero: \[ 400 - 40t = 0 \ 40t = 400 \ t = 10 \] Therefore, the vertex of the function occurs at \( (10, 0) \). This means that 10 seconds after starting, the car will be at your location (distance \( d = 0 \)).

3. Behavior of the Function:

   • For \( t < 10 \), the distance \( d \) is decreasing as the car approaches you.
   • For \( t > 10 \), the distance \( d \) increases again as the car moves past you.

4. Domain and Range:

   • Domain (t): Since time cannot take negative values in this context, the domain is \( t \geq 0 \).
   • Range (d): The distance can never be negative, so the range is \( d \geq 0 \).

5. Graph Behavior: The graph is a V-shaped function due to the absolute value, with the vertex at \( (10, 0) \) representing the point where the car meets you.

Conclusion of Key Features:

From the given options, the correct interpretation of the key features of the function \( d = |400 - 40t| \) is:

There is a vertex at (10, 0) which indicates that after 10 seconds the car is at the same location as you.

Correct answer choice: "There is a vertex at (10, 0) which indicates that after 10 seconds the car is at the same location as you."