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QUESTION ONE

A ball is thrown off a low cliff to the beach below. It is thrown into the air with an initial upward velocity of 5 meters per second. The following graph models its height above sea level in meters over time in seconds.

The graph image shows a dot at (0,10), sloping up to (0.5,11), and then going back down, hitting (2,0). The Y coordinate represents Height Above Sea Level (in Meters), and the X coordinate represents time (in seconds).

1A: Would a linear function, a quadratic function, or an exponential function be
most appropriate as a model for this graph? Explain your choice.

1B: Use the graph to create a table containing at least three points that you can
determine exactly from the graph.

1C: Use your table to create an equation that models this graph and scenario.

1D: Interpret the meaning of the y-intercept of this graph in terms of the given
context.

1E: Write your own question that asks a student to interpret a different key feature
of this graph in terms of the given context. Then, provide the solution for your
question.

1 answer

QUESTION ONE

1A:

A quadratic function would be the most appropriate model for this graph. This is because the motion of the ball can be described by a parabolic trajectory. The ball is thrown upwards with an initial velocity, reaches a maximum height, and then falls back down due to gravity. This pattern is characteristic of quadratic functions, which can model the height of an object in projectile motion under the influence of gravity.

1B:

From the given graph, we can determine the following points:

  1. (0, 10) — The height of the ball at time 0 seconds.
  2. (0.5, 11) — The height of the ball at time 0.5 seconds.
  3. (2, 0) — The height of the ball at time 2 seconds, indicating it has hit the beach.

1C:

To create a quadratic equation that models this graph, we can use the general form of a quadratic equation \( h(t) = at^2 + bt + c \), where \( h(t) \) is the height at time \( t \).

  1. We know \( c = 10 \) because the ball starts at a height of 10 meters at \( t = 0 \).

  2. At \( t = 0.5 \), \( h(0.5) = 11 \): \[ h(0.5) = a(0.5)^2 + b(0.5) + 10 = 11 \] \[ 0.25a + 0.5b + 10 = 11 \] \[ 0.25a + 0.5b = 1 \] (Equation 1)

  3. At \( t = 2 \), \( h(2) = 0 \): \[ h(2) = a(2)^2 + b(2) + 10 = 0 \] \[ 4a + 2b + 10 = 0 \] \[ 4a + 2b = -10 \] (Equation 2)

Now we have a system of equations:

  1. \( 0.25a + 0.5b = 1 \) (Equation 1)
  2. \( 4a + 2b = -10 \) (Equation 2)

To eliminate \( b \), we can multiply Equation 1 by 4: \[ a + 2b = 4 \] (Equation 3)

Now we can solve Equations 2 and 3:

  • From Equation 2: \( 2b = -10 - 4a \)
  • Substituting for \( b \) in Equation 3 gives us: \[ a + (-10 - 4a)/2 = 4 \] \[ 2a - 10 - 4a = 8 \] \[ -2a = 18 \] \[ a = -9 \]

Substituting \( a = -9 \) back to find \( b \): \[ 0.25(-9) + 0.5b = 1 \] \[ -2.25 + 0.5b = 1 \] \[ 0.5b = 3.25 \] \[ b = 6.5 \]

Putting it all together, the equation for the height of the ball over time is: \[ h(t) = -9t^2 + 6.5t + 10 \]

1D:

The y-intercept of the graph, which is at the point (0, 10), means that the initial height of the ball when it is thrown is 10 meters above sea level. This sets the starting point for the ball's motion, indicating that it is thrown from a height of 10 meters above the beach.

1E:

Question: What does the x-coordinate of the vertex of this parabola represent in the context of the ball's motion, and how could we determine its value?

Solution: The x-coordinate of the vertex of the parabola represents the time at which the ball reaches its maximum height. The vertex can be found using the formula \( t = -\frac{b}{2a} \). Substituting \( a = -9 \) and \( b = 6.5 \): \[ t = -\frac{6.5}{2(-9)} = \frac{6.5}{18} \approx 0.361 \text{ seconds} \] Thus, the ball reaches its maximum height approximately 0.361 seconds after it is thrown.