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docker41 [41]
3 years ago
12

A sinusoidal function whose period is π2 , maximum value is 10, and minimum value is −4 has a y-intercept of 10. what is the equ

ation of the function described?

Mathematics
2 answers:
valentina_108 [34]3 years ago
8 0

Answer:

The equation of the function is f(x)=7\sin(4x+\frac{\pi}{2})+3.

Step-by-step explanation:

The general form of sinusoidal function is

f(x)=a\sin(bx+c)+d .... (1)

where, a is amplitude, b is frequency, c is phase shift and d is vertical shift.

a=\frac{maximum-minimum}{2}=\frac{10-(-4)}{2}=7

b=\frac{2\pi}{period}=\frac{2\pi}{\frac{\pi}{2}}=4

d=\frac{maximum+minimum}{2}=\frac{10+(-4)}{2}=3

Substitute these values in equation (1).

f(x)=7\sin(4x+c)+3 ....(2)

It is given that the y-intercept of the function is 10. It means f(x)=10 at x=0.

10=7\sin(4(0)+c)+3

7=7\sin(c)

1=\sin(c)

\sin^{-1}(1)=c

c=\frac{\pi}{2}

Put this value in equation (2).

f(x)=7\sin(4x+\frac{\pi}{2})+3

Therefore the equation of the function is f(x)=7\sin(4x+\frac{\pi}{2})+3.

lions [1.4K]3 years ago
7 0
<span>the general equation of the sine curve is ⇒ y = a sin ( nx + α ) + b
</span><span>
</span><span>where : a is the amplitude
</span><span>
</span><span>             n = 2π/period
</span><span>
</span>             b = shift in the direction of y
             α°= shift in the direction of x

<span>Given period = π/2 ,
</span><span>
</span><span>maximum value is 10,
</span><span>
</span><span>minimum value is −4
</span><span>
</span><span>a y-intercept of 10.
</span>
 a = (maximum - minimum)/2 = (10 - -4)/2 = 7
 n = 2π/period = 2π/(π/2) = 4
 b = maximum - a = 10 - 7 = 3
to find α <span>⇒ y-intercept = 10
</span><span>
</span><span>y = 10 at x = 0
</span>
<span>substitute in the general function
</span><span>
</span>∴   y <span>= a sin ( nx + α ) + b
</span><span>
</span><span>∴ 10 = 7 sin ( 4*0 + α ) + 3
</span><span>
</span><span>∴ sin α = 1 ⇒⇒⇒ α = π/2</span>

So, <span>the equation of the function described is as attached in the figure
</span><span>
</span><span> y = 7 sin ( 4x + π/2 ) + 3
</span>

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