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Question
Math
Posted 6 months ago

The graph of a sinusoidal function has a minimum point at (0,3)(0,3) and then intersects its midline at (5π,5)(5 \pi, 5).

Write the formula of the function, where xx is entered in radians.
f(x)=
f(x)=

\square
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
Identify the characteristics of the sinusoidal function from the given points
step 2
The minimum point at (0,3)(0,3) indicates that the phase shift is 00 and the vertical shift is 33
step 3
The function intersects its midline at (5π,5)(5\pi, 5), which is one period away from the minimum. This tells us the midline is y=5y=5 and the amplitude is 22
step 4
Since the period is 5π5\pi, we can find the angular frequency, ω\omega, using the formula ω=2πperiod\omega = \frac{2\pi}{\text{period}}
step 5
Calculate the angular frequency: ω=2π5π=25\omega = \frac{2\pi}{5\pi} = \frac{2}{5}
step 6
Write the formula of the sinusoidal function using the amplitude AA, angular frequency ω\omega, phase shift DD, and vertical shift CC: f(x)=Asin(ωx+D)+Cf(x) = A\sin(\omega x + D) + C
step 7
Substitute the known values into the formula: f(x)=2sin(25x)+5f(x) = 2\sin\left(\frac{2}{5}x\right) + 5
Answer
f(x)=2sin(25x)+5f(x) = 2\sin\left(\frac{2}{5}x\right) + 5
Key Concept
Writing the formula of a sinusoidal function
Explanation
The formula of a sinusoidal function is derived from its amplitude, period (which gives the angular frequency), phase shift, and vertical shift. These values are determined from the characteristics of the function as described by its graph or given points.

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