Question

Jackson is swinging on a swing. His maximum height above the ground is 5 feet and his minimum height above the ground is 1 foot. What equations could model Jackson's height above the ground (y) after x seconds of swinging assuming that the x-axis represents the ground.

Answer 1

Jackson's height on a swing can be modeled using a sine or cosine function. Assuming a 2-second period, an example equation could be y = 2 sin(πx) + 3, where 2 is the amplitude of the swing and 3 is the midline (average height).

Step-by-step explanation:

Jackson's height (y) on a swing as a function of time (x) can be modeled by a trigonometric function, because swinging follows a periodic motion. We know that the maximum height is 5 feet and the minimum height is 1 foot. First, calculate the amplitude (A) which is half the difference between the maximum and minimum heights. The midline (D), which is the average of the maximum and minimum, also needs to be calculated. These are used in the standard sine or cosine function forms: y = A sin(Bx - C) + D or y = A cos(Bx - C) + D, where A is the amplitude, B affects the period of the function, C is the horizontal shift, and D is the vertical shift (midline).

In this case, A would be (5 - 1) / 2 = 2, and D would be (5 + 1) / 2 = 3. Without additional information regarding the period of the swing, we cannot definitively determine B. If we assume a period, for example 2 seconds, then B would be π since the period T of a sine or cosine function in this form is (2π) / B. There is also no horizontal shift, so C would be 0. Therefore, an example equation, assuming a 2-second period, could be: y = 2 sin(πx) + 3.

Answer 2

Jackson's swinging motion is best modeled with a sinusoidal function to represent the periodic nature of swinging, with parameters adjusted to match his maximum and minimum heights.

Step-by-step explanation:

Jackson's swinging motion on a swing can be modeled by trigonometric functions or a sinusoidal function because of the periodic nature of swinging. The equations for a sinusoidal function often have the form:

y = A * sin(B(x - C)) + D or y = A * cos(B(x - C)) + D,

where:

A is the amplitude (half of the difference between maximum and minimum heights),B affects the period (time it takes to complete one cycle),C shifts the graph horizontally (which shows the starting point in time when the motion starts), andD adjusts the vertical position (the average of the maximum and minimum heights).

In Jackson's case, we could use an equation like:

y = 2 * sin(B(x - C)) + 3,

This represents a swing with a 2-foot amplitude (difference between 5 feet and 1 foot is 4, so half of that is 2) and an average height of 3 feet (the midpoint between 1 foot and 5 feet). The values of B and C will depend on the time it takes for one complete back and forth swing and the point in time we consider as x = 0.