Mathematics » High School: Functions » Interpreting Functions

Understand the concept of a function and use function notation.

• 1. 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).

  Suggested tag: cc-9-12m-f-if-1

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

  Suggested tag: cc-9-12m-f-if-2

• 3. 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.

  Suggested tag: cc-9-12m-f-if-3

Interpret functions that arise in applications in terms of the context.

• 4. 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.^★

  Suggested tag: cc-9-12m-f-if-4

• 5. 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.^★

  Suggested tag: cc-9-12m-f-if-5

• 6. 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.^★

  Suggested tag: cc-9-12m-f-if-6

Analyze functions using different representations.

• 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.^★

  Suggested tag: cc-9-12m-f-if-7

  • a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

    Suggested tag: cc-9-12m-f-if-7a

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

    Suggested tag: cc-9-12m-f-if-7b

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

    Suggested tag: cc-9-12m-f-if-7c

  • d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

    Suggested tag: cc-9-12m-f-if-7d

  • e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

    Suggested tag: cc-9-12m-f-if-7e

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

  Suggested tag: cc-9-12m-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.

    Suggested tag: cc-9-12m-f-if-8a

  • 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.

    Suggested tag: cc-9-12m-f-if-8b

• 9. 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.

  Suggested tag: cc-9-12m-f-if-9