Summit Middle School classes for Andrew Busch
Andrew Busch - Summit
  • Home
  • Algebra 1
    • Alg 1B - Last Week
    • Alg1B 14 HW - Intro to Functions
    • Alg 1B 11 - Rational Expressions
    • Alg 1B 12 - Radical Expressions
    • Alg 1B 10 v2.0 - Quadratic Functions >
      • 10b Graphing with Pennies - Desmos Tutorial
      • 10i Snowboard Quadratic - Alg1B
      • 10 Quadratics Project
    • Alg 1B 10 Book - Factoring Quadratics
    • Alg 1B 9 - Exponential Functions
    • Alg 1B 8.5 - Representing Data
    • Alg 1B 13 - Inequalities
    • Alg 1B 8 - Best Fit Lines and Linear Regression
    • Alg 1B 7 - Linearity
  • Geometry
    • Geom Last Week
    • Geom 12 - Probability
    • Geom 11 - Circumference, Area, Volume
    • Geom 10-Circles
    • Geom 9 - Right Triangles and Trigonometry
    • Geom 8 - Similarity
    • Geom 7 - Quadrilaterals and Other Polygons
    • Geom 6 - Relationships Within Triangles
    • Geom 5 - Congruent Trianlges
    • Geom 4 - Transformations
    • Geometry 3.5 - Constructions
    • Geom 3 - Parallel and Perpendicular Lines
    • Geom 2 - Reasoning and Proofs
    • Geom 1 - Basics of Geometry
  • Programming
    • Directions for Sharing Programs with Me
    • Hour of Code
    • Intro to Python >
      • Installing and Using Portable Python
      • Introduction to Programming
      • Interactive Storyteller
      • Sophisticated Calculator
      • Getting Started with Games
      • Word Length Frequency
      • Substitution Cipher
      • Simple Game of Paddleball
      • Animating Many Objects
      • Accelerator
      • Applying Trigonometry
      • GIFs
      • Programmatic Art
      • Battleship
      • Pong
      • CodeCademy.com Suggested Work
      • Python Resources
    • Advanced Python >
      • Python Installation
      • Review of Intro to Programming
      • Objects and Classes >
        • More on Classes: Functions, Methods, Inheritance
        • Quadrilaterals
      • tkinter >
        • Paddle Ball
        • Light Bike
        • Frogger
        • Snake Game
        • Breakout
      • Reading and Writing Files
      • Directories and Importing Modules
      • Raspberry Pi
      • API's
      • Python Puzzles
  • Clubs
  • Graphing Calculator
  • PARCC Practice

Extra Practice Transforming cosine Functions

Picture
With amplitude (a), 
period (p), 
horizontal shift to the right (h), 
and vertical shift up (k).
Period:
The normal cosine function has a period of 2pi.

To find the period (p):
Find the distance between the tops of the waves. In this case, the distance is pi (~3.14...). So, the period is pi.
Picture

Amplitude
The amplitude is the measure of the radius of the corresponding circle.

To find the amplitude (a):

Measure the distance from the top to the bottom of the wave and divide by two. In the following graph, the distance from the top of the wave to the bottom of the wave is 4. So, the amplitude is 4/2 = 2.
Picture
Horizontal Shift

To find the horizontal shift (h):
Case 1: You can see the origin. If you can see the origin, find the horizontal distance from the origin. This is your best option when you are first starting out. When you throw in a period change and a horizontal shift, things can get tricky. The distance of the highest point from 0 is pi/4. So, in this case, h=pi/4
Picture
Case 2: You cannot see the origin AND there is not a period change.. Find some other point on the graph where you know where either the top or bottom of the wave is supposed to be. Remember, in the cosine function, cos(0) = 1, cos(pi/2) = 0, cos(pi) = -1, cos(3pi/2) = 0, and cos(2pi) = 1. In the graph below, just for fun let's look at the the bottom of the wave. It's measuring 5pi/4 instead of just pi. How far apart are those values?
5pi/4 - pi = 5pi/4 - 4pi/4 = pi/4. So, h = pi/4 (the same as case 1).
Picture


Vertical Shift
To find the vertical shift:
Take the average of the y-value for the top of the wave and the y-value for the bottom of the wave. In the graph above, the vertical shift is (1+-1)/2 = 0/2 = 0. In the graph below the vertical shift is (4+2)/2 = 6/2 = 3. Below, k = 3.
Picture



Take a moment to play around. 
To get the pi symbol in Desmos, just write "pi". It will automatically convert the value for you.

Practice:
For each of the following graphs, find each of the values: a, p, h, and k one at a time. Put them all together at the end to get the final function.
Graph 1:
Picture
Equation 1:
Picture
Graph 2:
Picture
Equation 2:
Picture
Graph 3:
Picture
f(x) = 2*cos((2pi/pi)(cosx - pi/3)) + 1
a = 2
p = pi
h = pi/3
k = 1
Graph 4:
Picture
a = 0.5
p = 2pi/3
h = pi/6
k = 0.5

Powered by Create your own unique website with customizable templates.