Alaska: Interpreting Functions Math Standards

UNDERSTAND THAT A FUNCTION FROM ONE SET (CALLED THE DOMAIN) TO ANOTHER SET (CALLED THE RANGE) ASSIGNS TO EACH ELEMENT OF THE DOMAIN EXACTLY ONE ELEMENT OF THE RANGE. IF F IS A FUNCTION AND X IS AN ELEMENT OF ITS DOMAIN, THEN F(X) DENOTES THE OUTPUT OF F CORRESPONDING TO THE INPUT X. THE GRAPH OF F IS THE GRAPH OF THE EQUATION 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. FOR EXAMPLE, THE FIBONACCI SEQUENCE IS DEFINED RECURSIVELY BY F(0) = F(1) = 1, F(N+1) = F(N) + F(N-1) FOR N ≥ 1.

FOR A FUNCTION THAT MODELS A RELATIONSHIP BETWEEN TWO QUANTITIES, INTERPRET KEY FEATURES OF GRAPHS AND TABLES IN TERMS OF THE QUANTITIES, AND SKETCH GRAPHS SHOWING KEY FEATURES GIVEN A VERBAL DESCRIPTION OF THE RELATIONSHIP. KEY FEATURES INCLUDE: INTERCEPTS; INTERVALS WHERE THE FUNCTION IS INCREASING, DECREASING, POSITIVE, OR NEGATIVE; RELATIVE MAXIMUMS AND MINIMUMS; SYMMETRIES; END BEHAVIOR; AND PERIODICITY.

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 NEGATIVE NUMBERS 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 FUNCTIONS EXPRESSED SYMBOLICALLY AND SHOW KEY FEATURES OF THE GRAPH, BY HAND IN SIMPLE CASES AND USING TECHNOLOGY FOR MORE COMPLICATED CASES.

• F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
• F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
• F-IF.7.c Graph polynomial functions, identifying zeros (using technology) or algebraic methods when suitable factorizations are available, and showing end behavior.
• F-IF.7.d (+) Graph rational functions, identifying zeros and discontinuities (asymptotes/holes) using technology, and algebraic methods when suitable factorizations are available, and showing end behavior.
• F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

WRITE A FUNCTION DEFINED BY AN EXPRESSION IN DIFFERENT BUT EQUIVALENT FORMS TO REVEAL AND EXPLAIN DIFFERENT PROPERTIES OF THE FUNCTION.

• F-IF.8.a 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.
• F-IF.8.b 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.

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.