How do you find the amplitude, period, and shift for #y = -5sin(x/2)#?

Question

Elijah Alexander · Accepted Answer

Amplitude #= |A| = |-5| = 5#

Period #" "=(2Pi)/(1/2) = 4Pi#

Shift #" "= C/B = 0/(1/2) = 0#

The Vertical Shift (D) = 0

Investigate the graph given below:

The General Form of the equation of the Cos function:

#color(green)(y = A*Sin(Bx + C) + D)#, where

A represents the Vertical Stretch Factor and its absolute value is the Amplitude.

B is used to find the Period (P):#" "P = (2Pi)/B#

C, if given, indicates that we have a place shift BUT it is NOT equal to #C#

The Place Shift is actually equal to #x# under certain special circumstances or conditions.

D represents Vertical Shift.

We observe that

Amplitude #= |A| = |-5| = 5#

Period #" "=(2Pi)/(1/2) = 4Pi#

Shift #" "= C/B = 0/(1/2) = 0#

The Vertical Shift (D) = 0

Hope this helps.

Carter Anderson · Answer

#5,4pi,0,0# #"the standard form of the sine function is "# #color(red)(bar(ul(|color(white)(2/2)color(black)(y=asin(bx+c)+d)color(white)(2/2)|)))# #"where amplitude "=|a|," period "=(2pi)/b# #"phase shift "=-c/b" and vertical shift "=d# #"here "a=-5,b=1/2,c=d=0# #"amplitude "=|-5|=5," period "=(2pi)/(1/2)=4pi# #"there is no phase / vertical shift"#

Delilah Aguilar · Answer

undefined To find the amplitude, period, and shift for the function $ y = -5\sin\left(\frac{x}{2}\right) $:

Amplitude: The amplitude of a sine function is the absolute value of the coefficient of the sine function. In this case, the amplitude is $ | -5 | = 5 $.

Period: The period of a sine function is determined by the coefficient of $ x $ inside the sine function. The period $ T $ is calculated as $ T = \frac{2\pi}{b} $, where $ b $ is the coefficient of $ x $. So, for $ y = -5\sin\left(\frac{x}{2}\right) $, the period $ T $ is $ \frac{2\pi}{\frac{1}{2}} = 4\pi $.

Shift: The shift of a sine function is any horizontal translation to the left or right. In the form $ y = A\sin(b(x - h)) + k $, $ h $ represents the horizontal shift. For $ y = -5\sin\left(\frac{x}{2}\right) $, there is no horizontal shift, so $ h = 0 $.