Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Graph exponential and logarithmic functions, showing intercepts and end behavior.
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Construct exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Interpret the parameters in an exponential function in terms of a context.