Question

What is the midline equation of y = -4sin (2x - 7) + 3?

Answer 1

`y = 3`.

Explanation:

The minimum and maximum value of
`\sin(2\, x - 7)` over all real
`x` are
`(-1)` and
`1`, respectively. Hence, the maximum and minimum value of
`y = -4\, \sin(2\, x - 7) + 3` would be:

• Maximum:
  `y = (-4)\, (-1) + 3 = 7`.
• Minimum:
  `y = (-4)\, (1)\ + 3 = (-1)`.

The midline equation of a sine wave is a horizontal line that is right in the middle of maximum and minimum
`y`-values of that sine wave. In the sine wave in this question, the average of the maximum and minimum
`y\!`-values is
`(1/2) \, (7 + (-1)) = 3`. Hence, the midline equation of this sine wave would be
`y = 3`.

Answer 2

`\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}`

Given:

▪
`\longrightarrow \sf{y = -4 \sin (2x - 7) + 3}`

This is the equation of the horizontal line equal to the average value of y for the function given.

`\leadsto` This will correspond to the value where the sine function has a value of 0, which is the average value of the function:

`\longrightarrow \sf{Y_(avg ) = −4(0) + 3 = 0 + 3 = 3}`

`\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}`

`\small\bm{The \: equation\: of \: the \: midline \: is \: \: y=3.}`