Secondary Math Amplitude, Midline, and Period of Waves

Size: px

Start display at page:

Download "Secondary Math Amplitude, Midline, and Period of Waves"

Lionel Barker
5 years ago
Views:

1 Secondary Math Amplitude, Midline, and Period of Waves

2 Warm UP Complete the unit circle from memory the best you can: 1. Fill in the degrees 2. Fill in the radians 3. Fill in the coordinates in the first quadrant Determine what coordinate is negative and positive in the next quadrants. Fill in the coordinates in the other three quadrants

3 What You will Learn How to extend the unit circle to the graph of the cosine and sine functions How to determine the amplitude, period, and midline of cosine and sine functions. How to write an equation for cosine or sine given a graph.

4 The Graphs of Sine and Cosine These are the graphs of =sin and =cos. We are going to study several things about them, but first we will answer the question, Where do they come from?

5 Imagine a flashlight shining a beam on the black spiral. The red curve is what the shadow of spiral would look like on the wall. The blue curve is what it would look like on the floor. This spiral comes from moving the unit circle as the angle rotates.

6 Here is another visual of how the unit circle produces waves.

7 Midline and Amplitude The midline of a wave is the center of the wave or the line that cuts the wave in half. The amplitude is the highest or lowest point of the wave measured from the midline.

8 Midline Continued The midline of a wave comes from adding a number to =sin or =cos. YOU WILL NEED TO DOWNLOAD DESMOS ONTO YOUR PHONE FOR THIS STEP Step 1: Type in sin()+. You can find sin under the FUNCTIONS/FUNCS menu. Step 2: Click the blue next to add slider: Step 3: Adjust as you like. How does it affect the graph? The line = is ALWAYS the midline of the wave.

9 Amplitude Continued The amplitude of a wave comes from multiplying a number by =sin or =cos. Step 1: Type in sin(). You can find sin under the FUNCTIONS/FUNCS menu. Step 2: Click the blue next to add slider: Step 3: Adjust as you like. How does it affect the graph? is ALWAYS the amplitude of the wave.

10 Example Find the amplitude and midline of the following waves: =2cos+3 =5sin 10 = 7cos+3.2

11 Finding Amplitude and Midline from a Graph What are the amplitude and midline for the graph at right? To find the midline of the wave, we average the highest -value and the lowest -value together (because the average is the middle ). The highest -value is 2 and the lowest - value is 6, so their average is () = 4. The midline is the line = 4. The amplitude is how far away the highest - value is from the midline. The highest -value is 2 which is 2 away from the midline. So the amplitude is 2.

12 Example Find the amplitude and midline of the wave.

13 The Period of a Wave The period of a wave is how long it takes for the wave to repeat itself. Using your unit circle, determine the period of sin andcos. (When do they start repeating?)

14 Changing the Period (Desmos) The period is changed by multiplying by a number. Step 1: Type in sin(!). You can find sin under the FUNCTIONS/FUNCS menu. Step 2: Click the blue! next to add slider: Step 3: Adjust! as you like. How does it affect the graph?! affects the period, but the period is ALWAYS " #.

15 Finding the Period from an Equation What is the period of =3cos4? The period % is ". % = " = " so the wave repeats # & itself every " units. Below is its graph.

16 Examples Find the amplitude, midline, and period of the following: = 1.5cos 3 1 =6sin

17 Homework #3 - Identify the midline, amplitude and period for the following function. ( =2sin ) +1

18 Finding the Period from a Graph Finding the period from a graph can be difficult because it is often a multiple of *, but if the graph is properly marked, it is easy to do. Below is the same wave with different markings. Let s find the period for both.

19 Examples Find the period of the following waves.

20 Is it a sine wave or a cosine wave? On Desmos, type in sin () for your first function and cos () for your second function. How are they similar? How are they different? =sin and =cos are the same wave, they are just shifted from each other! To decide whether to use sin or cos, we just need to look at the -axis. If the wave crosses the midline at the -axis, it is sin. If the wave has its highest or lowest point at the -axis, it is cos.

21 Example 1. Is it sine or cosine? 2. Find the midline. 3. Find the amplitude. 4. Find the period.

22 Writing Equations The general form of sine and cosine functions is as follows. =sin! + =cos! + Where is the amplitude is the midline value and " is the # period. You can determine the value of b by dividing 2π by the period. Write an equation for a sine wave with amplitude 10, midline = 6 and period " +.

23 Example Identify the midline, amplitude and period. Then write the equation for the function. ( = 3,-.+2

24 Example Identify the midline, amplitude and period. Then write the equation for the function. ( =/0,2 4

25 Example Identify the midline, amplitude and period. Then write the equation for the function. 1 = sin * 2

26 Homework #7 - Identify the midline, amplitude, and period. Then write the equation for the function. Use f(x) = sinx for the parent function.

27 Homework #11 - Identify the midline, amplitude, and period. Then write the equation for the function. Use f(x) = cosx for the parent function.

WARM UP. 1. Expand the expression (x 2 + 3) Factor the expression x 2 2x Find the roots of 4x 2 x + 1 by graphing.

WARM UP. 1. Expand the expression (x 2 + 3) Factor the expression x 2 2x Find the roots of 4x 2 x + 1 by graphing. WARM UP Monday, December 8, 2014 1. Expand the expression (x 2 + 3) 2 2. Factor the expression x 2 2x 8 3. Find the roots of 4x 2 x + 1 by graphing. 1 2 3 4 5 6 7 8 9 10 Objectives Distinguish between