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The graph of a sinusoidal function has a maximum point at

(
0
,
7
)
(0,7)left parenthesis, 0, comma, 7, right parenthesis and then intersects its midline at
(
3
,
3
)
(3,3)left parenthesis, 3, comma, 3, right parenthesis.
Write the formula of the function, where

xx is entered in radians.

(

)
=
f(x)=

User Jethar
by
8.8k points

1 Answer

6 votes

Answer:

f(x) = 4cos(πx/6) +3 or f(x) = 4sin(πx/6 +π/2) +3

Explanation:

You want a sinusoidal function that has a peak at (0, 7) and crosses the midline at (3, 3).

Amplitude

The amplitude of the function is the difference between the peak value and the midline value: 7 -3 = 4.

Vertical translation

The vertical translation of the function is the midline value.

Period

The x-value difference between the peak and the midline crossing is 1/4 period: (1/4)P = 3 - 0, so P = 12. The multiplier of x in the sinusoidal function argument is 2π/P = 2π/12 = π/6.

Horizontal translation

The peak of a cosine function is at x=0, so we can use a cosine function directly with no horizontal translation:

f(x) = 4cos(πx/6) +3 . . . . . . . . . amplitude 4, midline 3, period 12

If you insist on a sine function, then we can make use of the relation ...

cos(x) = sin(x +π/2)

and the function will be written ...

f(x) = 4sin(πx/6 +π/2) +3

__

Additional comment

The generic form of the function can be written as ...

f(x) = Asin(B(x+C))+D

where A = amplitude, B = 2π/period, C = left shift of rising midline crossing, D = vertical shift of midline

For a cosine function, C = left shift of positive peak.

The attached graph shows both the sine and cosine versions of the function.

<95141404393>

The graph of a sinusoidal function has a maximum point at ( 0 , 7 ) (0,7)left parenthesis-example-1
User DIGITALSQUAD
by
8.2k points
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