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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 equation of the function described?

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Matheng
the general equation of the sine curve is ⇒ y = a sin ( nx + α ) + b

where : a is the amplitude

             n = 2π/period

             b = shift in the direction of y
             α°= shift in the direction of x

Given period = π/2 ,

maximum value is 10,

minimum value is −4

a y-intercept of 10.

 a = (maximum - minimum)/2 = (10 - -4)/2 = 7
 n = 2π/period = 2π/(π/2) = 4
 b = maximum - a = 10 - 7 = 3
to find α ⇒ y-intercept = 10

y = 10 at x = 0

substitute in the general function

∴   y = a sin ( nx + α ) + b

∴ 10 = 7 sin ( 4*0 + α ) + 3

∴ sin α = 1 ⇒⇒⇒ α = π/2

So, the equation of the function described is as attached in the figure

 y = 7 sin ( 4x + π/2 ) + 3


Ver imagen Matheng

Answer:

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

Step-by-step explanation:

The general form of sinusoidal function is

[tex]f(x)=a\sin(bx+c)+d[/tex] .... (1)

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

[tex]a=\frac{maximum-minimum}{2}=\frac{10-(-4)}{2}=7[/tex]

[tex]b=\frac{2\pi}{period}=\frac{2\pi}{\frac{\pi}{2}}=4[/tex]

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

Substitute these values in equation (1).

[tex]f(x)=7\sin(4x+c)+3[/tex] ....(2)

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

[tex]10=7\sin(4(0)+c)+3[/tex]

[tex]7=7\sin(c)[/tex]

[tex]1=\sin(c)[/tex]

[tex]\sin^{-1}(1)=c[/tex]

[tex]c=\frac{\pi}{2}[/tex]

Put this value in equation (2).

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

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

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