write an equation of a sine graph with a phase shift right 3, a period of 5pi, a vertical translation down 6, and an amplitude of 3.

1 answer

To write the equation of a sine graph with the specified transformations, we can start with the general form of the sine function:

\[ y = a \sin(b(x - c)) + d \]

where:

  • \(a\) is the amplitude,
  • \(b\) determines the period,
  • \(c\) is the phase shift,
  • \(d\) is the vertical translation.

Given:

  • Amplitude (\(a\)): 3
  • Period: \(5\pi\)
  • Phase shift: right 3 (this means \(c = 3\))
  • Vertical translation: down 6 (this means \(d = -6\))

Step 1: Calculate \(b\) from the period

The period \(P\) of a sine function is given by the formula:

\[ P = \frac{2\pi}{b} \]

Setting the period to \(5\pi\):

\[ 5\pi = \frac{2\pi}{b} \]

To solve for \(b\), we rearrange the equation:

\[ b = \frac{2\pi}{5\pi} = \frac{2}{5} \]

Step 2: Assemble the equation

Now we can plug in our values for \(a\), \(b\), \(c\), and \(d\):

  • \(a = 3\)
  • \(b = \frac{2}{5}\)
  • \(c = 3\)
  • \(d = -6\)

Thus, the equation of the sine graph is:

\[ y = 3 \sin\left(\frac{2}{5}(x - 3)\right) - 6 \]

Final Equation

Putting it all together, we have:

\[ y = 3 \sin\left(\frac{2}{5}(x - 3)\right) - 6 \]

This is the equation of the sine graph with the specified transformations.