Question

How do you find period, amplitude, phase shift and midline of `f(x) = −4 sin(2x + π) − 5`?

Answer 1

please see below

Explanation:

we have standard form
`asin(bx+c)+-d`
`|a| " is amplitude," (2pi)/|b|" is period," " c is phase shift (or horizontal shift), d is vertical shift"`
comparing the equation with standard form
`a=-4,b=2,c=pi,d=-5`

midline is the line that runs between the maximum and minimum value(`i.e` amplitudes)
since the new amplitude is 4 and graph is shifted 5 units in negative `y-"axis"` (`d=-5`)

therefore maximum value is `4-5=-1` and minimum value is `-4-5=-9` midline will be centre of the region`(-1,-9)" " i.e" "-5`

phase change =`pi` units right(c is positive)
period is `(2pi)/2=pi`

it will be more clear from graph
graph{-4sin(2x+pi)-5 [-16.02, 16.01, -8.01, 8.01]}