Chapter 14 Day 8

Graphing Sin and Cos

•A periodic function is a function whose output values repeat at regular intervals. Such a function is said to have periodicity.

•The horizontal length of one cycle is called the period of the function.

•Trigonometric functions are periodic, therefore, their graphs repeat the same pattern over and over.

Graph Specific Forms of Sine and Cosine Functions

y =AsinB x−C( ) + D

y =AcosB x−C( ) + D

Amplitude of a Sine and Cosine Function

•The amplitude of a periodic function f is the distance from the midline to the maximum or minimum output value of the function.

•The constant factor A in acts as a scaling factor. It stretches or shrinks the graph vertically.▫If the , the curve is stretched▫If the , the curve is shrunk

A >1A <1

•What is the range of the function ?

 

•What is the range of the function ?

−1 ≤ y ≤ 1

−A ≤ y ≤ A

•The absolute value of A is called the amplitude of the function.

•The amplitude affects the range of a trig function.

•If A is negative, the function reflects over the x-axis.

Graphing the Sin Function

•What is the amplitude of the sine function?

 

•What is the period of the sine function?

•What is the domain of the sine function?

•What is the range of the sine function?

Graphing the Cos Function

•What is the amplitude of the cosine function?

 

•What is the period of the cosine function?

•What is the domain of the cosine function?

•What is the range of the cosine function?

Transformations of Sin and Cos Graphs

• The constant C in the equation acts as a “shifter” left or right. We won’t be shifting left or right, but we have done this for other functions.

• The constant D in the equation is also a “shifter” and it moves the graph up or down. This is the vertical displacement.

• The constant B in the equation changes the period of the function.

• The period of a sin or cos function is . • So, to find the period when B is not one, the

period of the function is the following:

y =AsinB x−C( ) + D y =AcosB x−C( ) + D

2π

2πB

Steps for Graphing•1. “Shift” the graph up or down and draw

in the new midline.

•2. Mark off the amplitude.

•3. Determine the period and scale your graph accordingly.

•4. Mark key points and sketch the graph. *Be careful if the A is negative, make sure to flip!*

Graph one period of each function. State the amplitude, period, maximum, and minimum.

1. y =2sinx

Graph one period of each function. State the amplitude, period, maximum, and minimum.

2. y =−4cosx

Graph one period of each function. State the amplitude, period, maximum, and minimum.

3. y =sin 4x( )

Graph one period of each function. State the amplitude, period, maximum, and minimum.

4. y =cos

12

x⎛

⎝⎜⎞

⎠⎟

Graph one period of each function. State the amplitude, period, maximum, and minimum.

5. y =−3sinx+ 3

Graph one period of each function. State the amplitude, period, maximum, and minimum.

6. y =2cos πx( )−3