Trig 2.4

Trig 2.4

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 $f\left(x\right)=3\sin\left(4x\right).$  

•  $\left|A\right|=3,\ T=4$  

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

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

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

A Note About Step 4 And Homework on Pearson

Since we are using Pearson for homework, you will not be expected to draw these graphs by hand! You will either be given multiple choice or a graphing tool.

-In the case of multiple choice, you may need obtain the 5 key points using the 4 steps in order to match the correct graph.

-If you are given a graphing tool, you will need the period, amplitude, and to identify any other transformations.

Multiple Choice

Quick Check 4! 

Determine the equation of the graph below.

• $-2\cos\left(\frac{\pi}{2}x\right)$

• $2\sin\left(\frac{\pi}{2}x\right)$

• $-2\sin\left(\pi x\right)$

• $-2\sin\left(\frac{\pi}{2}x\right)$