The height, [tex]$h$[/tex], in feet of a ball suspended from a spring as a function of time, [tex]$t$[/tex], in seconds can be modeled by the equation [tex]$h=3 \sin \left(\frac{\pi}{2}(t+2)\right)+5$[/tex]. Which of the following is the graph of this equation?
Answer (2)
The equation is h = 3 sin ( 2 π ( t + 2 ) ) + 5 .
Amplitude is 3, period is 4, phase shift is -2, and vertical shift is 5.
The graph oscillates between 2 and 8, has a period of 4, and is shifted 2 units to the left.
Identify the graph with these characteristics.
Explanation
Understanding the Problem We are given the equation h=3 ", "sin ", "left(", "frac{", "pi}{2}(t+2)\right)+5 which models the height, h , in feet of a ball suspended from a spring as a function of time, t , in seconds. We need to identify the graph of this equation.
Analyzing the Function Let's analyze the given sinusoidal function to determine its key features such as amplitude, period, phase shift, and vertical shift.
General Form of Sinusoidal Function The general form of a sinusoidal function is y = A sin ( B ( x − C )) + D , where:
∣ A ∣ is the amplitude
B 2 π is the period
C is the phase shift
D is the vertical shift
Identifying Key Features Comparing the given equation h = 3 sin ( 2 π ( t + 2 ) ) + 5 with the general form, we can identify the following:
Amplitude: A = 3
B = 2 π
Phase Shift: C = − 2 (since it's t + 2 )
Vertical Shift: D = 5
Calculating the Period Now, let's calculate the period: Period = B 2 π = 2 π 2 π = 2 π ⋅ π 2 = 4
Summarizing the Key Features So, we have:
Amplitude: 3
Period: 4
Phase Shift: -2
Vertical Shift: 5
This means the graph of the function has an amplitude of 3, a period of 4, a phase shift of -2 (shifted 2 units to the left), and a vertical shift of 5 (midline at h = 5).
Identifying the Graph Based on these characteristics, we can identify the correct graph. The graph should oscillate between 5 − 3 = 2 and 5 + 3 = 8 , have a period of 4, and be shifted 2 units to the left.
Examples
Sinusoidal functions are incredibly versatile and appear in many real-world scenarios. For instance, the motion of a pendulum, the height of tides, and the intensity of light can all be modeled using sinusoidal functions. Understanding the amplitude, period, phase shift, and vertical shift of these functions allows us to predict and analyze these phenomena accurately. By analyzing the equation, we can predict the maximum and minimum height of the ball, the time it takes to complete one full oscillation, and the starting position of the ball at time t=0.
The equation models the height of a ball with an amplitude of 3, a period of 4 seconds, a phase shift of -2 (2 units left), and a vertical shift of 5 (midline at 5 feet). The graph will oscillate between 2 and 8 feet. Look for a graph that meets these specifications.
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