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The graph of a sinusoidal function intersects its midline at [tex]\((0,2)\)[/tex] and then has a minimum point at [tex]\((3,-6)\)[/tex].

Write the formula of the function, where [tex]\(x\)[/tex] is entered in radians.

Sagot :

Semsee45idn

Answer:

[tex]f(x)=-8\sin\left(\dfrac{\pi}{6}x\right)+2[/tex]

Step-by-step explanation:

To write the equation for the graphed function, we can use the general form of the sine function:

[tex]f(x)=A\sin\left(B\left(x+C\right)\right)+D[/tex]

where:

  • |A| is the amplitude (distance between the midline and the peak). If A is negative, the graph is reflected across the x-axis.
  • 2π/B is the period (horizontal distance between consecutive peaks).
  • C is the phase shift (horizontal shift where negative is to the right).
  • D is the vertical shift (y = D is the midline).

The midline is the horizontal line (y = D) that the function oscillates around. Given that the function intersects its midline at (0, 2), the midline is y = 2, so:

[tex]D = 2[/tex]

The amplitude is the distance from the midline to the maximum or minimum value of the function. Given the minimum point (3, -6) and the midline y = 2, the amplitude is:

[tex]|A |= 2 - (-6) \\\\|A |=2+6\\\\|A |=8[/tex]

For the parent sine function y = sin(x), the midpoint between a minimum and the maximum peak occurs at x = 0. In the given function, the midpoint at x = 0 is between a maximum and the following minimum, indicating that the graph has been reflected across the x-axis. Therefore:

[tex]A = -8[/tex]

Additionally, as the midpoint between a maximum and minimum (or minimum and maximum) is at x = 0, there has been no horizontal shift from the standard position. Therefore:

[tex]C = 0[/tex]

For a sinusoidal function, the horizontal distance between the intersection of the midline and the minimum point is one-quarter of the period of the function. Therefore, to find the period T of the function, multiply the difference between the x-coordinates of the midline intersection point (x = 0) and the minimum point (x = 3) by 4:

[tex]T= 4 \cdot \left(3- 0\right) \\\\\\T= 4 \cdot \left(3 \right) \\\\\\T=12[/tex]

Since the period is equal to 2π/B, then:

[tex]\dfrac{2\pi}{B}=12 \\\\\\B=\dfrac{2\pi}{12}\\\\\\B=\dfrac{\pi}{6}[/tex]

Substitute the values of A, B, C and D into the general equation:

[tex]f(x)=-8\sin\left(\dfrac{\pi}{6}\left(x+0\right)\right)+2[/tex]

Simplify:

[tex]f(x)=-8\sin\left(\dfrac{\pi}{6}x\right)+2[/tex]

Therefore, the equation of the sinusoidal function, where x is in radians, is:

[tex]\Large\boxed{\boxed{f(x)=-8\sin\left(\dfrac{\pi}{6}x\right)+2}}[/tex]

View image Semsee45
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