Pre-Calculus Unit 1 Cheat Sheet Notes

Parent Functions and Transformations

• Key Features of Functions

  • Function Value
  • Domain/Range:
    • Domain: Values of X that the function can have
    • Range: Values of Y that the function can have
  • Increasing (Directly Proportional)
  • Decreasing (Indirectly Proportional)
  • Y-intercept: Where the function hits the y-axis, in the form (0, )
  • Zero/Root: Where the function hits the x-axis, in the form (, 0)

• Symmetry and Even/Odd Properties

  • Testing Symmetry:
    • x-axis: Replace y with -y
    • y-axis: Replace x with -x
    • Origin: Replace both x and y with their opposites
  • Even/Odd/Neither:
    • Even: Return original function
    • Odd: Return function with all signs changed
    • Neither: Any other result

Compositions

• Blob Method: Solving combined functions using a placeholder (blob) method
  • Copy down the first function until you hit a blob, then replace blob with the second function in parenthesis
• Real World Applications:
  • Create and solve equations from word problems using compositions

Inverses

• What is an Inverse?:
  • Passes the horizontal line test
  • Symmetric to the original function with respect to the line y=x
• Algebraic Method:
  1. Change the function name to y
  2. Switch y and x
  3. Solve for the new y
• Graphical Method:
  1. Create a table with function coordinate points
  2. Switch x and y in the coordinates
  3. Plot and connect the points
  4. Check symmetry with respect to y=x

Continuity, Limits, and End Behavior

• Continuity: Ability to draw the function without lifting the pencil

  • Limit equals the function at the point
  • Steps:
    1. Test limit from the left
    2. Test limit from the right
    3. Check if the limit exists
    4. Test function value
    5. Confirm the limit equals the function

• Limits: Not functions but values approaching a point

  • Limit exists if the left limit equals the right limit

• End Behavior: Behaviors at the graph's ends

  • Utilize limit notation to describe