The Student Council organized a potato throwing contest on the school field. Students used catapults to shoot the potatoes into the air and down the field. Some students launched the potatoes while other students were in charge of collecting data using a motion detector and a timer. Students analyzed the data to describe the path that each potato followed Team B launched their potato. According to the data collected they determined the function that modeled the potato’s path was 𝑦 = −(𝑥 − 15)^2 + 81 where x is the distance from the goal line in feet and y is the height of the potato in feet. Team C’s potato flew through the air and landed down the field. Using the collected data they calculated that the equation of the path of the potato through the air was 𝑦 = −0.75𝑥^2 + 18𝑥 −33, where x is the distance from the goal line in feet and y is the height of the potato in feet.

Question

The Student Council organized a potato throwing contest on the school field. Students used catapults to shoot the potatoes into the air and down the field. Some students launched the potatoes while other students were in charge of collecting data using a motion detector and a timer. Students analyzed the data to describe the path that each potato followed Team B launched their potato. According to the data collected they determined the function that modeled the potato’s path was 𝑦 = −(𝑥 − 15)^2 + 81 where x is the distance from the goal line in feet and y is the height of the potato in feet. Team C’s potato flew through the air and landed down the field. Using the collected data they calculated that the equation of the path of the potato through the air was 𝑦 = −0.75𝑥^2 + 18𝑥 −33, where x is the distance from the goal line in feet and y is the height of the potato in feet.

a) If a prize is given for maximum height, which team would be the winner? Explain how
you made your decision.

b) If a prize is given for the maximum distance travelled horizontally, which team would be
the winner? Explain how you made your decision

Answer 1

a) To determine the winner for the maximum height, we need to find the vertex of each function. The vertex of a parabola in standard form 𝑦 = 𝑎(𝑥 - ℎ)^2 + 𝑘 is (ℎ, 𝑘).

For Team B's function: 𝑦 = −(𝑥 − 15)^2 + 81, the vertex is (15, 81).
For Team C's function: 𝑦 = −0.75𝑥^2 + 18𝑥 − 33, we can find the vertex using the formula ℎ = -𝑏/(2𝑎). In this case, a = -0.75 and b = 18. Substituting these values into the formula, we get ℎ = -18/(2*(-0.75)) = -18/(-1.5) = 12. Therefore, the vertex is (12, 𝑦) where 𝑦 is the value of the function at 𝑥 = 12.

To determine the winner, we compare the heights (𝑦-values) of the two vertices. The vertex (15, 81) is higher than (12, 𝑦) because the y-coordinate of the vertex indicates the maximum height. Therefore, Team B is the winner for the maximum height.

b) To determine the winner for the maximum distance travelled horizontally, we need to find the x-intercepts of each function. The x-intercepts represent the distances to which the potatoes were launched.

For Team B's function: 𝑦 = −(𝑥 − 15)^2 + 81, to find the x-intercept, we set 𝑦 = 0 and solve for 𝑥:
0 = −(𝑥 − 15)^2 + 81
(𝑥 − 15)^2 = 81
Taking the square root of both sides:
𝑥 − 15 = ±9
𝑥 = 15 ± 9
Therefore, the x-intercepts are 15 + 9 and 15 - 9, which are 24 and 6 respectively.

For Team C's function: 𝑦 = −0.75𝑥^2 + 18𝑥 − 33, we need to solve for 𝑥 when 𝑦 = 0:
0 = −0.75𝑥^2 + 18𝑥 − 33
Multiplying the equation by 4 to remove the decimal:
0 = -3𝑥^2 + 72𝑥 - 132
Factoring out a common factor of -3:
0 = -3(𝑥^2 - 24𝑥 + 44)
0 = 𝑥^2 - 24𝑥 + 44
Using the quadratic formula: 𝑥 = (−𝑏 ± √(𝑏^2 − 4𝑎𝑐))/(2𝑎), where a = 1, b = -24, and c = 44:
𝑥 = (−(-24) ± √((-24)^2 − 4(1)(44)))/(2(1))
𝑥 = (24 ± √(576 - 176))/(2)
𝑥 = (24 ± √(400))/(2)
𝑥 = (24 ± 20)/(2)
Therefore, the x-intercepts are (24 + 20)/2 = 22 and (24 - 20)/2 = 2.

To determine the winner, we compare the distances (x-values) of the x-intercepts. The larger the distance, the further the potato travelled horizontally. The x-intercept 24 from the function 𝑦 = −(𝑥 − 15)^2 + 81 represents the further distance compared to the x-intercepts 22 and 2 from the function 𝑦 = −0.75𝑥^2 + 18𝑥 − 33. Therefore, Team B is the winner for the maximum distance travelled horizontally.