Algebra II End-of-Course Exam Content Standards—Core: Exponential Functions (Priority: 30%)
Successful students will be able to use tables, graphs, verbal statements and symbols to represent, analyze, model, and interpret graphs of exponential functions. While some facility with the properties of logarithms may be helpful it is not required on the core exam. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems.
X1. Exponential functions
a. Determine key characteristics of exponential functions and their graphs.
- Key characteristics include domain and range, intercepts, and end behavior.
Tasks related to this benchmark: Bighorn Sheep, Match That Function, Equal Salaries for Equal Work?
b. Represent exponential functions using tables, graphs, verbal statements, and equations. Represent exponential expressions in multiple forms. Translate among these representations.
- Know that exponential functions have the general form f(x) = abx + c, b > 0, and b ≠ 1.
- Distinguish between and graph exponential functions that are growth functions, such as f(x) - 3.2x where x > 0, or decay functions, such as f(x) - 3.2-x where x > 0.
- Decide if a given graph or table of values represents an exponential
function.
Be aware that it can be very difficult to distinguish exponential graphs from graphs of other functions, particularly polynomial functions, over small regions or particular subsets of their domains. Sometimes the context of an underlying situation can suggest a likely type of function model.
- Translate from exponential to logarithmic form and vice versa (e.g., ab - c → loga c - b)
c. Describe the effect that changes in a parameter of an exponential function have on the shape and position of its graph.
- Explain or illustrate the effect that changes in a parameter (a or c) or the base (b) have on the graph of the exponential function f(x) = abx + c.
Task related to this benchmark: Bighorn Sheep
d. Recognize, express, and solve problems that can be modeled using exponential functions, including those where logarithms provide an efficient method of solution. Interpret their solutions in terms of the context.
- A logarithm is an exponent that depends on the base used. Logarithms provide an efficient method for solving problems with variable exponents. Limitation: Logarithms may be used to solve problems but will not be used in the text of an item.
If xx = z, x > 0, x ≠ 1, and z > 0, then y is the logarithm to the base x of z. The equation y = logxz is one of three equivalent forms of expressing the relation xy = z (the other being
). - Exponential problems in which the student must determine the exponent may also be solved using graphing techniques.
- Exponential growth functions using base e may be included (e.g., A - Pert, or y - Cekt). Formulas will be provided in these cases.
- This includes using and interpreting appropriate units of measurement and precision for the given application.
Example: If a culture had 500 cells at noon and 600 cells at 1:00 PM, what is the approximate doubling time of the cell population? Approximately how many cells will there be at 4:00 PM?
Example: If radioactive iodine 123 has a half-life of 8 days, what percentage of an original dose will remain in a patient’s body twelve hours after a medical test has been performed in which iodine 123 was administered?
Task related to this benchmark: Bighorn Sheep