0 f(0)
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0 ?
π/4 ?
π/2 ?
3π/4 ?
π ?
use the table of values to graph f(0) = 4 sin(20) and find the amplitude, midline, and period
To graph the function f(0) = 4 sin(20), we can start by determining the amplitude, midline, and period.
The general form of the sine function is f(x) = A sin(Bx + C) + D, where A represents the amplitude, B determines the period (T = 2π/B), C is the phase shift, and D is the vertical shift.
In this case, f(0) = 4 sin(20), so we have A = 4, B = 20, C = 0, and D is not mentioned, which means there is no vertical shift. Therefore, D = 0.
Amplitude:
The amplitude represents the maximum distance the graph extends above and below the midline. In this case, the amplitude is A = 4.
Midline:
The midline is the horizontal line around which the graph oscillates. It is given by the equation y = D, which in this case is y = 0. Thus, the midline is the x-axis.
Period:
The period of a sine function is the distance between two consecutive peaks (or troughs) of the graph. It is determined by the value of B in the general form of the sine function. In this case, we have B = 20, so the period is T = 2π/B = 2π/20 = π/10.
Now, let's fill in the table of values using the given x-values: 0, π/4, π/2, 3π/4, and π.
0 f(0) = 4 sin(20) = 4 * sin(0) = 0
π/4 f(π/4) = 4 sin(20 * π/4) = 4 * sin(5) ≈ 3.225
π/2 f(π/2) = 4 sin(20 * π/2) = 4 * sin(10) ≈ 3.082
3π/4 f(3π/4) = 4 sin(20 * 3π/4) = 4 * sin(15) ≈ 0.989
π f(π) = 4 sin(20 * π) = 4 * sin(20) ≈ -1.298
Now we can graph the function f(0) = 4 sin(20) with the given x-values and corresponding y-values:
(π/10, 4)
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0 π/4 π/2 3π/4 π
The graph is a sinusoidal curve that oscillates between 4 and -4 with a period of π/10. The midline is the x-axis, and the amplitude is 4.
Note: Since there is no vertical shift (D = 0), the midline coincides with the x-axis.