AP Precalculus Unit 3 Exp/Logs

• Arithmetic Sequence


  is a sequence that has successive terms that share a common difference. d
• The nth term of an arithmetic sequence can be found using the formula An = A1 +d (n-1).
• Geometric Sequence


  a sequence that has successive terms that share a common ratio. r
• The nth term of a geometric sequence can be found using this formula gn =gk(r)n-k.
• Arithmetic sequences behave like ....


  Linear functions
• Geometric sequences behave like....


  Exponential functions
• What reasoning should you give when determining an arithmetic sequence?


  Because over equal length input values the output values change at a constant rate.
• What reasoning should you give when determining a geometric function?


  Because over equal length values the output values change proportionally.
• An exponential function is an equation raised to an exponential variable
• What is the difference between Exponential Growth and Exponential Decay? 


  Exponential Growth = b>1

  Exponential Decay = 0<b<1

  
• Limit Notation for Exponential Functions 


  limabx=oo

  x-> +/- oo

  limabx=-oo

  x->+/-oo

  limabx=0

  x->=+/-oo
• Increasing vs Decreasing Exponential Functions


  Exponential Functions are ALWAYS increasing or decreasing. They never switch from one to another, so they have NO relative extrema

  
• Concavity: the state of being concave up/down


  Concave Up will mostly be seen ABOVE the x-axis

  Concave down will mostly be seen BELOW the x-axis

  
• Exponential Rules/ Properties(Reminder)

  *If and only if indicated a - sign within the exponent don't automatically assume its a negative shift like 8^x-1= 1/8^x to which can also be like this 8^x/3=8^1/3 (x)
• Transformation Rules Of Functions 


  (f(x)=b^x+h)=stretch/shrink

  
• Residuals ▪ The residuals are the values of the dependent variable that are left over after the independent variable has been subtracted from the dependent variable.
• Residual Plotting: A plotting technique that uses the residuals to determine the best fit line.
• How to determine if a residual is over or underestimated?


  Check the sign of the residual: positive means underestimated, negative means overestimated.
• How would you know if a model for a given data is appropriate?

  The residual plot shows NO PATTERN
• Exponential Function Manipulation 


  When multiplication = addition and subtraction = division
• When it comes to manipulating Exponential functions...


  DON'T change the constant UNLESS it has an exponent
• Power Property


  Multiply both the exponent WITHIN and OUTSIDE the parenthesis

  
• Product Property 


  When multiplying two Exponential functions with the same base ADD the exponents

  
• Quotient Property


  When dividing two Exponential functions with the same base SUBTRACT the exponents

  
• Negative Exponent Property


  When dealing with a negative exponent ALWAYS shift it downward

  
• What are the two types of transformations mentioned?


  Vertical and horizontal transformations
• Where do vertical transformations occur in a function?


  Outside the parenthesis
• What does the addition or subtraction symbol indicate in vertical transformations?


  It determines up or down movement
• How is a stretch or shrink represented in vertical transformations?


  As a factor of the number being multiplied in FRONT of the function
• Where do horizontal transformations occur in a function?


  Inside the parenthesis
• What does the addition or subtraction symbol indicate in horizontal transformations?


  It determines right or left movement
• How is a stretch or shrink represented in horizontal transformations?

  As a factor of $\frac{1}{\text{number}}$ being multiplied from the INSIDE of the function
• What is the reflection over the x-axis represented as?

  • f(x)
• What is the reflection over the y-axis represented as?


  f(-x)
• What are the key characteristics of vertical transformations?


  • Occur outside the parenthesis
  • Addition/subtraction indicates up/down movement
  • Stretch/shrink is a factor of the number multiplied in front of the function
• What are the key characteristics of horizontal transformations?


  • Occur inside the parenthesis
  • Addition/subtraction indicates right/left movement
  • Stretch/shrink is a factor of $\frac{1}{\text{number}}$ multiplied from inside the function
• What are the rules for reflections in transformations?

  • Reflection over the x-axis: -f(x)
  • Reflection over the y-axis: f(-x)
• Modeling Exponential Functions

  • When given a data

  table with selected values from several Exponential functions use successive difference method to find the potential growth pattern. to then write an equation to model the function

  • However when given a table with only 2 values and plug them into the graphing calculator and use the indicated regression model to form an equation