Question

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

Answer 1

Answer:

$y = 3$.

Step-by-step 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:

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

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

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