My experience tossing the object and finding the three pairs of coordinates was quite straightforward. I followed the instructions given and was able to mark the spots accurately.
The three pairs of coordinates I obtained are as follows:
Starting point: (0, 0)
Top of the arc: (2, 4)
Final resting point: (5, 0)
To write the equation of a quadratic function that models the data, I can use the standard form of a quadratic equation: y = ax^2 + bx + c. Since the object follows a parabolic path, the equation of its trajectory will be quadratic.
First, I can substitute the coordinates of the starting point into the equation. Since the starting point is (0, 0), we know that when x = 0, y = 0. Therefore, c = 0.
Next, I can substitute the coordinates of the top of the arc into the equation. We have (2, 4), which means when x = 2, y = 4. Substituting these values into the equation, we get:
4 = a(2)^2 + b(2)
4 = 4a + 2b
Finally, I can substitute the coordinates of the final resting point into the equation. We have (5, 0), which means when x = 5, y = 0. Substituting these values into the equation, we get:
0 = a(5)^2 + b(5)
0 = 25a + 5b
Now we have a system of two equations:
4 = 4a + 2b
0 = 25a + 5b
Solving this system of equations, we find that a = -4/15 and b = 8/3.
Therefore, the equation of the quadratic function that models the data is:
y = (-4/15)x^2 + (8/3)x
The main challenge I had was making sure the measurements were accurate, especially when marking the spot where the object reached the top of the arc. It required quick reflexes to mark the spot before the object continued its descent. However, with some practice, it became easier to get the measurements more accurately.
For this discussion you will need tape, a ruler or yardstick, and a small object that will not bounce, such as a bottle cap or an eraser. Mark a spot at the bottom of a wall with a small piece of tape. Starting at this piece of tape, throw the small object up and forward so that it travels in an arc and brushes the wall. Use your other hand to quickly mark the spot on the wall where the object reached the top of the arc. Put a small piece of tape at this spot. Put a third piece of tape at the bottom of the wall where the object came to rest.
Let the tape at the starting point be the origin. Use the ruler or yardstick to measure the horizontal and vertical distances from the origin to the tape at the top of the arc. Record the coordinates. Use the ruler or yardstick to measure the horizontal distance from the origin to the final resting point. Record the coordinates. You now have three pairs of coordinates from the path of the object.
Describe your experience tossing the object and finding the three pairs of coordinates. List the three pairs of coordinates. Then use the coordinates to write the equation of a quadratic function that models the data. Show your work and comment on any challenges that you had.
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