Chapter 4: Circular Functions

Lesson 7: Scale-Change Images to Trig Functions

Mrs. Parziale

• Parent Sine or Cosine Function:

• General form of Sine or Cosine Function: siny x cosy x

siny a bx cosy a bx

Example 1:

• Using your TI83, Graph and Compare the following functions.

• How do these two graphs compare?– What are their maximum points?– What are their minimum points?– What relationship do you see with the equation?– What is the amplitude?

sin 3siny x y x

2 4 4x y

a

What’s the Amplitude?

• What is the amplitude of the following equations?

• a)

• b)

• c)

4siny x1sin2

y x

2siny x

Example 2:

• Using your TI83, Graph and Compare the following functions.

• Describe how the two graph are different.– What is the value of the maximum?– What is the value of the minimum?– How many cycles of the sine curve are there on each

graph from 0 to 2?– What is the Period?

sin sin 2y x y x

2

b

How Many Cycles?

• How many cycles of the sine curve are on the graphs for the equations below?

a)

b)

c)

sin 3y x

1sin2

y x

sin 4y x

Example 3:

• Identify the amplitude and period for the following sine functions.

5siny xamplitude: ____

period: _______

sin 4y x

amplitude: ____

period: _______

Example 3, cont.:

• Identify the amplitude and period for the following sine functions.

amplitude: ____

period: _______

3sin 2y x

Graphing Sine and Cosine Functions:

• To graph the sine and cosine functions, you need five key points which include the x- and y-intercepts and the maximum and minimum points. Take the following steps when graphing the sine and cosine functions.

• Step 1: Find the amplitude and period of the function.• Step 2: Divide the section of the graph into four equal

parts.• Step 3: Find the intercept points.• Step 4: Find the max and min points.• Step 5: Graph the five points and draw a smooth curve

through them.

Example 4:

• Graph • Amplitude = ________• Period = __________• x-intercepts = ______, _____, • y-intercept = ______• max = _________ • min = _________

sin 4y x

Example 5:

• Graph • Amplitude = ________• Period = __________• x-intercepts = ______, _____, • y-intercept = ______• max = _________ • min = _________

3sin 2y x

Example 6:

• Graph • Amplitude = ________• Period = __________• x-intercepts = ______, _____, • y-intercept = ______• max = _________ • min = _________

cos 2y x

Example 7:

• Graph • Amplitude = ________• Period = __________• x-intercepts = ______, _____, • y-intercept = ______• max = _________ • min = _________

.25sin 4y x

Graphing the Tangent Function

• The parent tangent function has a period of . When graphing the tangent function, you will need to know the x-intercept, vertical asymptotes, the halfway points, and the period.

• Period = ___________ • Vertical asymptotes are defined at odd

multiples of ______________

Example 8:

• Graph .• Period = _______ • x-intercept = _______• Vertical

Asymptotes = ________

• Halfway points = ___, ___

3tan 2y x

Closure

• Given the equation• What is the amplitude?• Explain how an amplitude of ½ affects the

graph compared to the parent function?• How many cycles appear within 2π?• What is the period of the curve?• How does a period of affect the

graph?

1sin32

y x

2

3