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

Question

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

Use this as an example of how to write your answer.
"For the coordinates, I got (0,0) (2,4) (4,0). Using the general form of y=ax^2 +bx + c, we can substitute the numbers in y = -x^2 + 4x"

Answer 1

When I tossed the object and found the three pairs of coordinates, I had to ensure that I accurately marked the spot on the wall where the object reached the top of the arc and where it came to rest. It required some coordination between throwing the object and quickly marking the spots on the wall.

The three pairs of coordinates I obtained were:

Starting point: (0,0)
Top of the arc: (2,4) (horizontally 2 units away from the origin and vertically 4 units above the origin)
Final resting point: (4,0) (horizontally 4 units away from the origin and at the same vertical level as the origin)

To write the equation of a quadratic function that models the data, I can use the vertex form of a quadratic equation: y = a(x-h)^2 + k, where (h,k) represents the vertex.

To find the vertex, I can use the coordinates of the top of the arc since it represents the highest point of the parabolic path. The vertex will be the midpoint between the origin and the top of the arc. In this case, the midpoint is ((0+2)/2, (0+4)/2) = (1, 2).

Using the vertex (h,k) = (1, 2), I can substitute these values into the vertex form equation:

y = a(x-1)^2 + 2

Now I need to find the value of 'a' in order to complete the equation. For that, I will substitute one of the points not on the vertex, such as the final resting point (4,0), into the equation:

0 = a(4-1)^2 + 2
0 = a(3)^2 + 2
0 = 9a + 2
-2 = 9a
a = -2/9

Therefore, the equation that models the data is:

y = (-2/9)(x-1)^2 + 2

The main challenge I encountered during this exercise was accurately marking the spots on the wall, especially the exact point where the object reached the top of the arc. It required some quick hand-eye coordination to ensure precise measurements.