Sketch a graph of the function [tex]f(x)=2 \sin \left(\frac{\pi}{2} x\right)-3[/tex]
Answer (2)
Identify the function as a sinusoidal function of the form f ( x ) = A sin ( B x − C ) + D .
Determine the amplitude ∣ A ∣ = 2 , period B 2 π = 4 , and vertical shift D = − 3 .
Calculate key points on the graph over one period using x = 0 , 1 , 2 , 3 , 4 .
Sketch the sinusoidal curve based on the amplitude, period, vertical shift, and key points. Graph sketched as described
Explanation
Analyze the Function We are asked to sketch the graph of the function f ( x ) = 2 sin ( 2 π x ) − 3 . This is a sinusoidal function, specifically a sine function that has been transformed. To sketch the graph, we need to identify the key features such as amplitude, period, and vertical shift.
Identify Key Parameters The general form of a sinusoidal function is f ( x ) = A sin ( B x − C ) + D , where:
∣ A ∣ is the amplitude.
B affects the period, and the period is given by ∣ B ∣ 2 π .
C is the phase shift.
D is the vertical shift.
In our case, f ( x ) = 2 sin ( 2 π x ) − 3 , so A = 2 , B = 2 π , C = 0 , and D = − 3 .
Calculate Amplitude, Period, and Vertical Shift
Amplitude: The amplitude is ∣ A ∣ = ∣2∣ = 2 .
Period: The period is ∣ B ∣ 2 π = 2 π 2 π = 2 π ⋅ π 2 = 4 .
Vertical Shift: The vertical shift is D = − 3 . This means the midline of the function is at y = − 3 .
Find Key Points Now, let's find some key points on the graph over one period. We'll use the standard sine function's key points (0, 2 π , π , 2 3 π , 2 π ) and adjust them according to our period and vertical shift.
x = 0 : f ( 0 ) = 2 sin ( 2 π ( 0 )) − 3 = 2 sin ( 0 ) − 3 = 0 − 3 = − 3
x = 1 : f ( 1 ) = 2 sin ( 2 π ( 1 )) − 3 = 2 sin ( 2 π ) − 3 = 2 ( 1 ) − 3 = − 1
x = 2 : f ( 2 ) = 2 sin ( 2 π ( 2 )) − 3 = 2 sin ( π ) − 3 = 0 − 3 = − 3
x = 3 : f ( 3 ) = 2 sin ( 2 π ( 3 )) − 3 = 2 sin ( 2 3 π ) − 3 = 2 ( − 1 ) − 3 = − 5
x = 4 : $f(4) = 2\sin(\frac{\pi}{2}(4)) - 3 = 2\sin(2\pi) - 3 = 0 - 3 = -3
Plot Points and Sketch the Curve So, we have the following points:
(0, -3)
(1, -1)
(2, -3)
(3, -5)
(4, -3)
These points represent one full period of the function. We can plot these points and sketch the sinusoidal curve. The curve oscillates between -5 and -1, with a midline at y = -3.
Final Description of the Graph The graph of f ( x ) = 2 sin ( 2 π x ) − 3 is a sine wave with an amplitude of 2, a period of 4, and a vertical shift of -3. It oscillates between -5 and -1.
Examples
Understanding sinusoidal functions is crucial in many fields. For instance, in electrical engineering, alternating current (AC) voltage and current are often modeled as sine waves. The amplitude represents the peak voltage or current, the period determines the frequency of the AC signal, and the phase shift indicates the timing difference between voltage and current. By analyzing these parameters, engineers can design and optimize electrical circuits and systems effectively.
The graph of f ( x ) = 2 sin ( 2 π x ) − 3 features an amplitude of 2, a period of 4, and is vertically shifted downward by 3. Key points include (0, -3), (1, -1), (2, -3), (3, -5), and (4, -3). The curve oscillates between -5 and -1 around the midline at y = -3.
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