Evidence of Understanding

• compare and contrast exponential functions with other functions that contain exponents
  • describe similarities and differences between power and exponential models
    • compare graphs of power functions and exponential functions (focus on y = x² and y = 2^x, y = x³ and y = 3^x, and y =√x and y = ½^x)
    • possible extension: notice similarities between x² and 2^x and use that to anticipate and describe possible similarities between √x and the inverse of 2^x
  • generate and justify a function rule from a table, sequence, graph, or situation
    • determine when a power, exponential, or neither function model is appropriate
      
• investigate and transform expressions containing exponents into equivalent forms
  • rewrite expressions using exponent properties a^m ● aⁿ = a^{m + n} (product), (a^m)ⁿ = a^mn (power), a⁰ = 1 (zero),a^m/aⁿ = a^{m -  n}(quotient), and a^-m = 1/a^m
    • explore and describe the commutativity of exponent rules
    • prove expressions with different bases are equivalent (Example: 64³ = 8⁶ = 4⁹ = 2¹⁸)
  • explain the relationship between a rational exponent and its equivalent radical form
    • connect the definition of a square root or cube root to its equivalent exponent
  • convert between expressions with radicals and expressions with rational exponents
    
• analyze and evaluate expressions or equations using properties of exponents
  • identify equations or expressions that can be rewritten with common bases to solve problems
    • Example: solve 3^2x = 27^{x + 1} by rewriting as 3^2x = (3³)^{x + 1}or solve (1/16)^x+ 13 = 77 by simplifying and rewriting as (2^-4)^x = 2⁶
  • evaluate expressions and solve equations using radicals or rational exponents
    • Example: Evaluate 81^3/2 or 7√(128⁴) or Solve (5x - 2)^5/3 - 1 = 31
    • apply properties of exponents and common bases to evaluate without a calculator
    • use rational exponents to determine an unknown base rate

Develop conceptual understanding:

power function, exponential function, exponent properties, commutative, base, exponent, rational exponent, radical form, square root, cube root  

Supporting terms to communicate:

function, dependent, independent, input, output, domain, range, integer, rational number, equation, expression