Interpreting Functions

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. ketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.

Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, and x is the input (an element of the domain), then f(x) is the output (an element of the range) Graphically, the graph is y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers(Generally, the scope of high school math defines this subset as the set of natural numbers 1, 2, 3, 4, ) By graphing or calculating terms, students should be able to show how the recursive sequence a1 = 7, an = an 1 + 2; the sequence sn = 2(n 1) + 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context).

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior

Graph exponential and logarithmic functions, showing intercepts and end behavior

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms.

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 and decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.