Bungee Jumper Quadratic Function: Height, Domain, Range, and Vertex Analysis

• Domain: The domain of a function refers to all possible input values (in this case, time ).
  • Physically, time cannot be negative, so .
  • The jump ends when the bungee jumper touches the river, which means . We will calculate this time later, but for now, let’s call it . So, the jumper is in the air from until .
  • To find , set :
    
    
    
    
    
  • Therefore, the physical domain is or approximately .
  • Physical Significance of Domain: The domain represents the time interval during which the bungee jumper is in the air, from the moment of the jump ( ) until they reach the river surface ( ).
• Range: The range of a function refers to all possible output values (in this case, height ).
  • The maximum height is the initial height since and the jumper immediately starts falling. So, the maximum height is meters.
  • The minimum height is when the jumper touches the river, which is 0 meters.
  • Therefore, the physical range is .
  • Physical Significance of Range: The range represents all possible heights of the bungee jumper above the river surface, from the river level (0 meters) up to the initial height of the jump (210 meters).

• t-coordinate of the vertex:
  
• h-coordinate of the vertex:
  Substitute into the function:
  
• Vertex:
• What the vertex represents in this scenario: The vertex represents the initial moment of the jump ( seconds) and the initial height of the jumper (210 meters). Since the initial velocity is 0, this is also the maximum height the jumper reaches.

• As determined from the vertex calculation, the maximum height occurs at seconds.
• The maximum height is meters.
• Explanation using the formula and scenario:
  The height function is . This is a parabola opening downwards because the coefficient of (which is -0.5) is negative. For such a parabola, the vertex represents the maximum point.
  The t-coordinate of the vertex is given by . In our case, , so . This means the maximum height is reached at seconds.
  Substituting into the height function gives meters.
  In this specific scenario, the initial velocity is 0 m/s, meaning the jumper starts from rest. Therefore, the highest point they will ever be is their starting point, which is at seconds and a height of 210 meters. From this point, they immediately begin to fall.

• Height after 20 seconds: The height after 20 seconds of the jump is 10 meters.
• What this situation represents: This means that 20 seconds after the jump, the bungee jumper is 10 meters above the river surface. This is a point during their descent, just before they would hit the river (which we found to be at approximately 20.49 seconds).

• Vertex:
• t-intercept:
• Another point (e.g., from part (e)):
• Another point: . So, .

      ^ h (meters)
      |
  210 + . (0, 210) - Vertex/Initial Height
      |   \
      |     \
  160 +       . (10, 160)
      |         \
      |           \
   10 +             . (20, 10)
    0 +-------------------------------------> t (seconds)
      0         10        20   (20.49, 0)

• Increasing: The height function is increasing when the graph is going upwards from left to right. In this scenario, the vertex is at , which is the starting point and the maximum height. Since the parabola opens downwards, the height never increases after . If we consider the mathematical function without physical constraints, it would be increasing for . However, physically, time starts at . So, there is no interval where the height is increasing for .
• Decreasing: The height function is decreasing when the graph is going downwards from left to right. From the vertex at , the graph continuously goes downwards until it hits the t-axis.
  Therefore, the height is decreasing for the time interval or approximately .

• Axis of Symmetry: For a parabola , the axis of symmetry is the vertical line .
  In our case, .
  So, the axis of symmetry is the line (which is the h-axis).
• Explanation using the axis of symmetry:
  The axis of symmetry passes through the vertex of the parabola. In this scenario, it represents the exact moment the bungee jumper begins their fall. Since the initial velocity is zero, the jumper is at their highest point at . The symmetry implies that if the jumper had been launched upwards from a lower point to reach 210m at , their path before would be a mirror image of their path after . However, in this specific physical scenario, the jump starts at , so we only consider the right side of the axis of symmetry. The axis of symmetry marks the point in time where the jumper’s vertical motion changes direction (or in this case, starts its downward motion from rest).

• h-intercept: This occurs when .
  .
  The h-intercept is .
  Representation: This represents the initial height of the bungee jumper above the river surface at the moment the jump begins ( seconds).
• t-intercept: This occurs when .
  
  
  . (We discard the negative root as time cannot be negative).
  The t-intercept is or approximately .
  Representation: This represents the time it takes for the bungee jumper to reach the river surface (height = 0 meters) after the jump.

• Domain: seconds, or approximately seconds.
• Physical Significance of Domain: The domain represents the duration of the bungee jump, from the initial moment of the jump until the jumper touches the river.
• Range: meters.
• Physical Significance of Range: The range represents all possible heights of the bungee jumper above the river surface during the jump, from the river level to the initial height.

• Vertex: .
• Representation: The vertex represents the initial position of the bungee jumper at the start of the jump ( seconds) and their maximum height (210 meters) above the river.

• The bungee jumper reaches maximum height at seconds.
• The maximum height is meters.
• Explanation: The function is a downward-opening parabola. Its vertex, which corresponds to the maximum point, occurs at . Since , the vertex is at . At this time, the height is meters. Given the initial velocity is 0, the jumper starts at their highest point and immediately begins to fall.

• The bungee jumper reaches the height of 11m at approximately seconds.

• The height after 20 seconds is meters.
• Representation: This situation represents the bungee jumper being 10 meters above the river surface 20 seconds into the jump, indicating they are still in the air and descending.

• The bungee jumper will touch the river at approximately seconds.

• Increasing: The height is not increasing for .
• Decreasing: The height is decreasing for the time interval seconds, or approximately seconds.

• Axis of Symmetry: The line (the h-axis).
• Explanation: The axis of symmetry passes through the vertex , which is the starting point and maximum height of the jump. It signifies the initial moment when the jumper begins their descent from rest. Due to the physical constraints of the scenario (time cannot be negative), we only observe the behavior of the function to the right of this axis.

• h-intercept: .
  • Representation: This is the initial height of the bungee jumper (210 meters) at the start of the jump ( ).
• t-intercept: or approximately .
  • Representation: This is the time (approximately 20.49 seconds) when the bungee jumper reaches the river surface (height = 0 meters).