Transformations of Sine and Cosine

Graphs of the Sine and Cosine Functinos

Recall Transformations From Algebra

We will use transformations of functions typically covered in an advanced algebra course (these can also be reviewed in chapter 1.6 in this text) as one method to graph trigonometric functions. 

Multiple Choice

Recall Check! 

Use the basic function  $$f\left(x\right)=x^2$$  to obtain the equation of a graph of a new function  $$g\left(x\right)$$ using the following transformations: 
Reflect the graph across the  x-axis
Vertically stretch the graph by a factor of 2
Shift the graph 4 units to the right
Shift the graph 1 unit up.

• $$g\left(x\right)=2\left(-x-4\right)^2+1$$  

• $$g\left(x\right)=-2\left(x+4\right)^2+1$$  

• $$g\left(x\right)=-\frac{1}{2}\left(x-4\right)^2+1$$  

• $$g\left(x\right)=-2\left(x-4\right)^2+1$$  

Multiple Select

Quick Check 1!
Describe in words the transformations of the basic function  $$f\left(x\right)=\sin x$$ used to obtain the graph of $$g\left(x\right)=-\sin\left(2x\right).$$  Select all that apply.

• Shift the graph of  $$f\left(x\right)$$   2 units to the right

• Reflect the graph of  $$f\left(x\right)$$   across the x-axis 

• Vertically stretch the graph of $$f\left(x\right)$$  by a factor of 2.

• Reflect the graph of  $$f\left(x\right)$$  across the y-axis

• Horizontal compression of  $$f\left(x\right)$$   by a factor of  $$\frac{1}{2}$$  .

Multiple Choice

Quick Check 2!
Obtain the graph of $$f\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)$$  using transformations.

Do you notice anything special about the graph of  $$f\left(x\right)?$$  

Multiple Choice

Quick Check 3!
Determine the period and amplitude of $$f\left(x\right)=3\sin\left(4x\right).$$  

• $$\left|A\right|=3,\ T=4$$  

• $$\left|A\right|=\frac{1}{3},\ T=\frac{\pi}{2}$$  

• $$\left|A\right|=3,\ T=\frac{\pi}{2}$$  

• $$\left|A\right|=\frac{1}{3},\ T=4$$  

Multiple Choice

Quick Check 4! 

Determine the equation of the graph below.

• $$-2\cos\left(\frac{\pi}{2}x\right)$$

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

• $$-2\sin\left(\pi x\right)$$

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

Practice Time

Multiple Choice

Sine or Cosine?

• y=sinx
• y=cosx

Multiple Choice

What is the amplitude of

y = -3sin (7x) -2

• 7

• -2

• 3

• -3

Multiple Choice

What is the period of either graph? 
y = sin(x)      &   y = cos (x)

• Pi
• 2Pi
• Pi/2
• Pi/4 

Multiple Choice

What is the period?

• 4π

• 2π

• π

• 8π

Multiple Choice

What is the equation of the graph?

• y = 2 cos x

• y = sin x

• y = 2 sin x

• y = sin 2x

Multiple Choice

What is the equation of graph?

• y = sin 5x

• y = cos 5x

• y = 5 sin x

• y = 5 cos x

Multiple Choice

What is the equation of the graph?

• y = -2 cos x - 3

• y = 3 cos x - 2

• y = -7 cos x - 3

• y = -3 cos x - 4

Multiple Choice

What is the equation of this graph?

• y=cos2x
• y=2sin2x
• y=2cos2x
• y=2sinx

Multiple Choice

What is the vertical shift for y = 4cos(¹/₂x +π) - 2?

• up 2

• left 2

• right 2

• down 2

Multiple Choice

Name the function and how many units the graph shifted vertically.

• sine, 2 units up

• cosine, 2 units down

• cosine, 2 units up

• sine, 2 units up

Multiple Choice

Write an equation for the cosine function with amplitude 3, period 2pi, phase shift of pi, and vertical shift of 5.

• y = 5cos(x - pi) + 3

• y = 3cos(x - pi) + 5

• y = 3cos(2x - pi) + 5

• y = 5cos(2x - pi) + 3

Multiple Choice

What is the equation of the graph shown?

• y = -sinθ -1

• y = -sinθ/2 -1

• y = -2sin2θ + 1

• y = sinθ/2 + 1