Answered

At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Ask your questions and receive accurate, in-depth answers from our knowledgeable community members.

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
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.