Question

Math

Posted 6 months ago

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The graph of a sinusoidal function has a minimum point at $(0,3)$ and then intersects its midline at $(5 \pi, 5)$.
Write the formula of the function, where $x$ is entered in radians.
$f(x)=$
$\square$
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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 $(0,3)$ indicates that the phase shift is $0$ and the vertical shift is $3$

step 3

The function intersects its midline at $(5\pi, 5)$, which is one period away from the minimum. This tells us the midline is $y=5$ and the amplitude is $2$

step 4

Since the period is $5\pi$, we can find the angular frequency, $\omega$, using the formula $\omega = \frac{2\pi}{\text{period}}$

step 5

Calculate the angular frequency: $\omega = \frac{2\pi}{5\pi} = \frac{2}{5}$

step 6

Write the formula of the sinusoidal function using the amplitude $A$, angular frequency $\omega$, phase shift $D$, and vertical shift $C$: $f(x) = A\sin(\omega x + D) + C$

step 7

Substitute the known values into the formula: $f(x) = 2\sin\left(\frac{2}{5}x\right) + 5$

Answer

$f(x) = 2\sin\left(\frac{2}{5}x\right) + 5$

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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