AP Precalculus 2025: Sinusoidal Modeling of Guitar String Vibrations

Physical Context and Modeling of Guitar String Vibration

The physical phenomenon described involves the vibration of guitar strings, which move in a repetitive up-and-down or back-and-forth motion. This mechanical movement is mathematically represented using a periodic function, specifically a sinusoidal function denoted as $h(t)$. The function $h$ tracks the vertical displacement of a specific point, labeled Point X, on the string relative to its resting position. The resting position serves as the equilibrium point, or the midline of the sinusoidal graph. The unit of measurement for time $t$ is seconds, while displacement $h(t)$ is measured in millimeters ($mm$). According to the model's conventions, a positive value of $h(t)$ indicates that Point X is above the resting position, whereas a negative value indicates it is below the resting position.

Initial Conditions and Motion Characteristics

The motion of Point X at the start of the observation period ($t = 0$) is defined by its highest position, which is $2\,mm$ above the resting position. The sequence of motion follows a specific periodic cycle: starting at the maximum height of $2\,mm$, Point X moves downward through the resting position ($0\,mm$) until it reaches its lowest position, which is documented as $2\,mm$ below the resting position (represented as $-2\,mm$). Following this minimum, Point X moves back upward through the resting position to return to its initial maximum height of $2\,mm$. This complete cycle, from peak to peak, occurs at a high frequency. The transcript specifies that this complete motion occurs $200$ times in $1$ second. This frequency allows for the calculation of the period ($P$) of the function, which is the time required for one full oscillation:

$P = \frac{1}{200}\,s = 0.005\,s$

Analysis of Specific Coordinates for Points F, G, J, K, and P

The graph of function $h$ displays two full cycles of this vibration. Five distinct points are identified on the first cycle of the wave: F, G, J, K, and P. Based on the mechanical description of the vibration starting at the highest position at $t = 0$, we can determine the coordinates $(t, h(t))$ for each point by dividing the period ($0.005\,s$) into quarters.

Point F is the starting point at the maximum displacement: $F = (0, 2)$

Point G is the first instance where the string passes through the resting position while moving downward. This occurs at one-quarter of the period: $t = \frac{1}{4} \times 0.005 = 0.00125\,s$$G = (0.00125, 0)$

Point J represents the lowest position of Point X, occurring at one-half of the period: $t = \frac{1}{2} \times 0.005 = 0.0025\,s$$J = (0.0025, -2)$

Point K is the second instance where the string passes through the resting position, this time moving upward. This occurs at three-quarters of the period: $t = \frac{3}{4} \times 0.005 = 0.00375\,s$$K = (0.00375, 0)$

Point P marks the completion of the first full cycle, returning to the highest position: $t = 0.005\,s$$P = (0.005, 2)$

Mathematical Parameters for the Sinusoidal Equation

The function $h$ can be expressed in the general form $h(t) = a \sin(b(t+c)) + d$. To find the values of the constants $a$, $b$, $c$, and $d$, we evaluate the properties of the vibration:

1. Midline ($d$): The resting position is the average of the maximum ($2\,mm$) and minimum ($-2\,mm$) heights. $d = \frac{2 + (-2)}{2} = 0$

2. Amplitude ($a$): The amplitude is the distance from the midline to the peak. $a = 2 - 0 = 2$

3. Period and Angular Frequency ($b$): The period is $0.005\,s$. The value of $b$ is determined by the formula $b = \frac{2\pi}{P}$. $b = \frac{2\pi}{0.005} = 400\pi$

4. Phase Shift ($c$): At $t = 0$, the function is at its maximum. For a sine function without a shift ($c = 0$), the value at $t = 0$ is the midline ($0$). Since the function is at its peak at $t = 0$, it behaves like a cosine function. To represent this as a sine function, we use the identity $\sin(x + \frac{\pi}{2}) = \cos(x)$. Inside the argument $400\pi(t + c)$, we need the phase shift to result in $\frac{\pi}{2}$ when $t = 0$. $400\pi(c) = \frac{\pi}{2}$$c = \frac{1}{800}$

Thus, the function is $h(t) = 2 \sin(400\pi(t + \frac{1}{800}))$

Interval Analysis of Graph Behavior and Concavity

Part C focuses on the specific interval $(t_1, t_2)$, where $t_1$ is the t-coordinate of point G and $t_2$ is the t-coordinate of point J. Point G is $(0.00125, 0)$ and Point J is $(0.0025, -2)$.

In the interval $(t_1, t_2)$, the function $h$ is negative and decreasing. At point G, the string begins to move below the resting position, making $h(t) < 0$. It proceeds from $0\,mm$ toward the minimum of $-2\,mm$, which indicates a decrease in value. Therefore, the correct selection for the behavior of $h$ on this interval is option (d): "$h$ is negative and decreasing."

Regarding the concavity and rate of change on $(t_1, t_2)$, the graph is concave up. Beginning at the inflection point G, the slope of the curve is at its steepest negative value. As time progresses toward the relative minimum at point J, the slope becomes less negative, eventually reaching zero at point J. Because the slope (rate of change) is transitioning from a large negative value toward zero, the rate of change is increasing. This increasing rate of change mathematically corresponds to a graph that is concave up.