Functions by Forest

Functions by Forest

1. Absolute Value

   • Critical point is the vertex, (h,k)
   • represented by y=a|x-h|+k
     • a affects steepness
       • h affects horizontal shift
         • k affects vertical shift
   • Real World Applications
     • Distances for some purposes -- like graphing the distance away from home while traveling to some place
     • Banking Security -- the amount of effort put into securing an account increases as both the presence or lack of money increases

2. Quadratic

   • Ways to Write Quadratic Functions
     • Standard Form
       • f(x)=ax^2+bx+c
     • Vertex Form
       • f(x)=a(x-h)^2+k (h,k)=vertex
     • Intercept Form
       • f(x)=a(x-r1)(x-r2) (r1,0) (r2,0)=x-intercepts
   • Real World Appications
     • To Model Objects' Height WhileTraveling Through Air, Like a Kicked Soccer Ball or a Model Rocket
     • To Model How Much Profit is Made Over a Variable to find the Highest Point of Profit
   • Critical point is the vertex, (h,k)

3. Square Root

   • Written as f(x) = a (x - h)^1/2 + k
     • Domain=-h
     • Range= [k,∞) or (-∞,k]
     • a affects steepness
       • h affects horizontal shift
         • k affects vertical shift
   • Real World Applications
     • Suppose you are looking for the dimensions to a certain square area or an area that can be broken into square -- square root functions can help
     • 5^1/2 is seen a lot in nature -- from the distribution to tree branches to the distribution of sunflower seeds in a head
     • Electrical engineers use it in calculating power factor and electrical load distribution
       • This is also where i is handy, in electricity i must be used to go lower than the domain of 0 in square root functions

4. Cube Root

   • Written as f(x) = a (x - h)^1/3 + k
     • a affects steepness and vertical stretch
       • h affects horizontal shift
         • k affects vertical shift
     • Critical point=(h,k) -- the point of inflection
   • In addition to a domain of all real numbers, cubed root functions have a range of all real numbers.
   • They are easiest to graph when using perfect cubes for x values
   • Real World Applications
     • Determining the Edge of a Cube with a Given Volume
     • The Derivation of Geostationary Altitude
       • Using qualities of a given satallite orbiting in a perfect sphere, it's altitude can be calculated using a cubed root function

5. Exponential

   • Exponential Functions have the Parent Function of y=b^x and have a b>0 and a b≠0
     • Two Special Kinds of Exponential Functions when a>0 and r>0
       • Exponential Growth
         • y=a(1+r)^x
       • Exponential Decay
         • y=a(1-r)^x
       • The quantity 1+r is the growth factor and 1-r is the decay factor
     • They can be transformed using y=ab^x-h +k
       • h affects horizontal shift
         • k affects vertical shift
           • b affects changes in steepness
             • a affects horizontal shift
     • Exponentials are Special Because they Have an Asymptote
       • asymptote=k
   • Real World Applications
     • Used A LOT for Compounding Interest
       • uses A=P(1+r)^t
         • A=Amount of money you have after t years
           • P=Principal -- amount you initially invest or borrow
             • t=time of investment or loan in years
               • r=interest rate expressed as a decimal
                 • n-# of times compounded each year
       • Can Calculate Depreciation Using the Same Rules with a t<0

6. Domain = All Real Numbers

7. Commonly Used for Money

8. by Forest Clevenger, #303, Day 14, for tatromath R3