graphing sin & cos

graphing sin & cos

by Christine Lennox

​y=sinx

​-We just learned why y=sinx looks like this.

-Now we see what happens when we put an "a" in front and see what happens when we graph y=asinx

​Amplitude (a)

• ​The amplitude is half the distance from the max to the min

• ​Think of it as how far the max is from the "midline"

• ​Amplitude is always positive, thus amplitude = |a|

• ​The red graph has an amplitude of 1

• ​The blue graph has an amplitude of 2

Multiple Choice

What is the amplitude of the graph shown?

• 6

• $$\pi$$  

• 3

• 1/3

• not given

Multiple Choice

What is the amplitude of the graph shown?

• 2

• $$\pi$$  

• 8

• 4

• not given

​y=k+sinx or y= sin(x)+k

• ​What happens if we add/subtract a number?

• ​The k is the "midline"

• ​Where can we draw the midline, this is our k

• ​k shifts the graph up (+k) and down (-k)

Multiple Choice

What k value does this graph have?

• 1

• -1

• 2

• -2

Multiple Choice

What k value does this graph have?

• 1

• -1

• 2

• -2

​Let's combine the two,

y=k+asinx

• ​What is the a?

• ​What is the k?

Multiple Choice

Given the graph, which are the correct values of a and k in y=k+asinx?

• a=-2

  k=.2

• a=.5

  k=-2

• a=1

  k=-2

• a=.5

  k= $$\pi$$  

​y=sin(x-h)

• ​What does the h do?

• ​The red graph is y=sinx

• ​The blue graph is

​y=sin(x-h)

• ​If h is positive then it shifts right example: y=sin(x-pi)

• ​If h is negative then it shifts left

  example: y=sin(x+pi)

• ​The key is to highlight one "S" in the sin curve, when you have the starting point, that gives you the h

• ​There are multiple options!!!

Multiple Choice

Look at the graph, instead of the starting point for a +sinx curve being at (0,0), where did it move?

• Left pi/2

• Left pi

• Right pi/2

• Right pi

Multiple Choice

Now that we agree, the graph shifted right pi, what is the equation that would yield/give the graph shown?

• $$y=\sin\left(x+\pi\right)$$  

• $$y=\sin\left(x-\pi\right)$$  

• $$y=\pi\sin x$$  

• $$y=-\pi\sin x$$  

Multiple Choice

Which other equation would yield/give the graph shown?

• $$y=\sin\left(x+2\pi\right)$$  

• $$y=\sin\left(x-\frac{\pi}{2}\right)$$  

• $$y=\sin\left(x+\frac{\pi}{2}\right)$$  

• $$y=\sin\left(x+3\pi\right)$$  

​y=sinbx

• ​What happens when we change the b?

• ​The blue graph is y = sin x

• ​The red graph is y = sin bx

• ​Look for one cycle of sin, this is called the "period"

• ​Bigger b's give smaller periods

• ​Smaller b's give bigger periods

Multiple Choice

What is the period? (How long does it take to make one cycle?)

• $$\frac{\pi}{2}$$  

• $$\pi$$  

• $$\frac{\pi}{4}$$  

• $$2\pi$$  

Multiple Choice

What is the period? (How long does it take to make one cycle?)

• $$\frac{\pi}{2}$$  

• $$\pi$$  

• $$0.5$$  

• $$\frac{3\pi}{2}$$  

Multiple Choice

What is the period? (How long does it take to make one cycle?)

• $$2$$  

• $$2\pi$$  

• $$3\pi$$  

• $$4\pi$$  

​what is the b given the period?

• ​Now that we can find the period. we can calculate the b

• ​period = pi

• ​

​

​

​

• ​so b = 2

• ​

Multiple Choice

If the period is  $$4\pi$$  , what is the b?

$$period\ =\ \frac{2\pi}{b}$$  

• 4

• 2

• $$\frac{1}{2}$$  

• 1

Multiple Choice

If the period is  $$2$$  , what is the b?

$$period\ =\ \frac{2\pi}{b}$$  

• $$\pi$$  

• 2

• $$2\pi$$  

• 1

Multiple Choice

Given the equation, what is the period?

$$y=2\sin4x$$  

• $$\frac{\pi}{4}$$  

• $$\frac{\pi}{2}$$  

• $$\pi$$  

• 1

Multiple Choice

Given the equation, what is the period?

$$y=\frac{1}{2}\sin\pi x$$  

• $$4\pi$$  

• $$\pi$$  

• 2

• 4

​Combining b and h

• ​y = sin b(x-h) or y = a sin (bx-c)

• If we get y = sin b(x-h) we can instantly determine the period and shift

• ​If you get y = sin (bx-c), we can instantly determine the period BUT we have to factor out the b to get y=sinb(x-h) in order to see the shift!