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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)
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 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.

The function is an increasing function, so as the time increases, the distance the car is from you decreases at a constant rate.

1 answer

To analyze the function \( d = |400 - 40t| \), let's break down its components and key features:

  1. Understanding the Function:

    • The expression \( 400 - 40t \) represents the distance from the starting point (400 feet) minus the distance traveled by the car (40 feet per second times time \( t \)).
    • The absolute value indicates that distance cannot be negative.

  2. Finding the Key Features:

    • Vertex: The vertex occurs when \( 400 - 40t = 0 \). This happens when: \[ 400 = 40t \ t = 10 \] At \( t = 10 \) seconds, the distance \( d = 0 \), meaning the car is at your location.

  3. Domain and Range:

    • Domain: The time \( t \) can start from \( 0 \) (the moment the car starts moving) and can take any non-negative value: \( t \geq 0 \).
    • Range: Since distance cannot be negative, the range will be \( d \geq 0 \). The car starts at 400 feet away, so it approaches you but never goes below 0.

  4. Behavior of the Function:

    • The function decreases as \( t \) increases, until it reaches 0 at \( t = 10 \) seconds. After that, it starts increasing again. This indicates the car has passed you (at t > 10) and is now moving away.

Key Features Summary:

  • The vertex at \( (10, 0) \) indicates that the car is at the same location as you after 10 seconds.
  • The function decreases to 0 at \( t = 10 \) and then increases afterward.
  • The domain is \( t \geq 0 \) (the car cannot travel back in time), and the range is \( d \geq 0 \) (distance cannot be negative).

Conclusion:

The option that best fits 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.

Incorrect Options:

  • The options claiming the domain is restricted to non-positive values and the range allowing negative values are incorrect as outlined above.
  • The function is indeed decreasing up until \( t = 10 \) seconds, but does not represent an increasing function for the entire duration \( t \). It decreases to 0 and then increases, so the statement about it being an increasing function overall is misleading.

Thus, the correct answer is the one that identifies the vertex at \( (10, 0) \).