Understanding the Relationship: Amplitude & Vertical Dilation of Sine/Cosine, Schemes and Mind Maps of Trigonometry

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Amplitude and Vertical Dilation of Sine and Cosine Functions

In this activity, you will discover the relationship between the vertical dilation A and the amplitude of: (^) y A sin( x ) and (^) y A cos( x )

Applet : Amplitude_Vertical_Dilation

Terminology: Vertical dilation stretches or compresses the function along the y-axis.

Before you begin, check that: A=1 and that the checkbox “Allow Fractions for A” is unchecked.

Questions and Answers:

1. Write down the parent functions of y A sin( x )and y A cos( x ). What is the value of A

and what is the amplitude of these parent functions?

Answer: Parent functions are y = sin(x) and to y = cos(x), respectively. In both parent

functions, the value of A is 1 and the amplitude =1.

2. Move slider A in any direction (Aт0). What type of transformation does this represent?

Answer: Moving A (Aт0) in any direction gives a vertical dilation or (since |A|>1), the transformation is a vertical stretch.

3. What happens to y = Asin(x) and to y = Acos(x) when A increases in the positive direction?

Answer: When A increases in the positive direction and since A>1, the functions stretch

vertically more and more.

4. Set A = 2. Write the equations for both the sine and cosine functions below.

Answer: The equations are y = 2sin(x) and y = 2cos(x).

5. What is the amplitude of both sine and cosine under this vertical dilation?

Answer: The amplitudes of y = 2sin(x) and y = 2cos(x) is amplitude =2.

6. Set A = 4. Write the equations for both the sine and cosine functions below.

Answer: The equations are y = 4sin(x) and y = 4cos(x).

7. What is the amplitude of both sine and cosine under this vertical dilation?

Answer: The amplitudes of y = 4sin(x) and y = 4cos(x) is amplitude =4.

There seems to be a relationship between the amplitude of a sinusoidal function and A.

Let's see if you can see the pattern.

Table 1: Positive Whole Numbers for A.

Dilation Factor A = 1 A = 2 A = 4

Amplitude amplitude = 1 amplitude = 2 amplitude = 4

8. Write an equation for the amplitude in terms of A when A>0.

Answer: The amplitude of the functions y = Asin(x) and to y = Acos(x) is amplitude =A.

Note: Because we are not looking at A<0, at this time they will not write the correct equation.

9. Review question: Set A=1 and then A=-1. What changes in the graph of the sine function? What changes in the cosine function? Complete the equations: -sin(x)=? and -cos(x)=?

Answer: Correct answers for the sine function are: (a) The graphs of y = sin(x) and y = sin(-x) are

reflections across the x-axis or (b) The graphs of y = sin(x) and y = sin(-x) are reflections across

the y-axis. Correct answer for the cosine function are (a) The graphs of y = cos(x) and y = cos(-x)

are the same. sin(-x)=-sin(x) (odd function) and cos(-x)=cos(x) (even function).

10. Let’s see what happens when A is negative? Set A = –1. What is the amplitude of the sine and cosine function? What is the amplitude of the sine and cosine function when A = –2 and when A = –4? Fill in row2 of Table 2 below.

Answer: The amplitude of y = sin(-x) is amplitude =1 (the function is not dilated, just inverted).

The amplitude of y = cos(-x) is amplitude =1 (the function is not dilated, nor inverted). The

amplitudes of y = sin(-2x) and y = cos(-2x) is amplitude =2 and the amplitudes of y = sin(-4x) and

y = cos(-4x) is amplitude =4.

11. Does the amplitude change between when A is positive and A is negative?

Answer: No.

18. Let’s look at A between -1 and 0. Complete Table 4 with A=-1, A=-½ and A=-¼. Check that row2 and row3 are the same.

Table 4: Negative Fractions for A Dilation Factor A = -1 A = -½ A = -¼ Amplitude amplitude = 1 amplitude = ½ amplitude = ¼ Amplitude using Equation amplitude =|A| =|-1|=

amplitude =|A| =|-½|=½

amplitude =|A| =|-¼|=¼

Conclusions (cross out incorrect responses in blocks and fill in blanks):

1. Changing the value of A dilates (stretches or compresses) the functions y = Asin(x) and

y = Acos(x) along the horizontal vertical axis.

2. When |A|>1, the functions y = Asin(x) and y = Acos(x) are stretched compressed and

the amplitude A of these functions is amplitude > 1 amplitude < 1.

3. When the amplitude < 1 of the functions y = Asin(x) and y = Acos(x), this means that

|A|>1 0<|A|<1 and the functions are stretched compressed vertically.

4. The amplitude of y = Asin(x) and y = Acos(x) is determined by the formula

Amplitude = |A|