Functions

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Standards

• CCSS.MATH.CONTENT.HSF.IF.C.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.
• CCSS.MATH.CONTENT.6.SP.B.5 — Summarize numerical data sets in relation to their context, such as by:
• CCSS.MATH.CONTENT.8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• CCSS.MATH.CONTENT.HSF.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
• CCSS.MATH.CONTENT.HSF.BF.B.4.B — (+) Verify by composition that one function is the inverse of another.
• CCSS.MATH.CONTENT.HSF.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
• CCSS.MATH.CONTENT.HSF.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
• CCSS.MATH.CONTENT.HSF.LE.A.1.A — Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
• CCSS.MATH.CONTENT.HSF.IF.C.7.C — Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
• CCSS.MATH.CONTENT.HSF.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
• CCSS.MATH.CONTENT.HSF.IF.B.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.
• CCSS.MATH.CONTENT.HSF.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.
• CCSS.MATH.CONTENT.HSF.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
• CCSS.MATH.CONTENT.HSF.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions.
• CCSS.MATH.CONTENT.HSF.LE.A.1.B — Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
• CCSS.MATH.CONTENT.HSF.IF.C.7.D — (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
• CCSS.MATH.CONTENT.HSF.BF.B.4 — Find inverse functions.
• CCSS.MATH.CONTENT.HSF.IF.A.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).
• CCSS.MATH.CONTENT.HSF.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
• CCSS.MATH.CONTENT.8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
• CCSS.MATH.CONTENT.HSF.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
• CCSS.MATH.CONTENT.HSF.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
• CCSS.MATH.CONTENT.8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
• CCSS.MATH.CONTENT.HSF.BF.A.1.C — (+) Compose functions.
• CCSS.MATH.CONTENT.HSF.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.
• CCSS.MATH.CONTENT.HSF.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• CCSS.MATH.CONTENT.HSF.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
• CCSS.MATH.CONTENT.HSF.BF.B.5 — (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
• CCSS.MATH.CONTENT.HSF.BF.A.1.A — Determine an explicit expression, a recursive process, or steps for calculation from a context.
• CCSS.MATH.CONTENT.HSF.BF.B.4.D — (+) Produce an invertible function from a non-invertible function by restricting the domain.
• CCSS.MATH.CONTENT.HSF.BF.A.1.B — Combine standard function types using arithmetic operations.
• CCSS.MATH.CONTENT.HSF.BF.B.4.A — Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
• CCSS.MATH.CONTENT.HSF.LE.A.4 — 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.
• CCSS.MATH.CONTENT.HSF.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
• CCSS.MATH.CONTENT.8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
• CCSS.MATH.CONTENT.HSF.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
• CCSS.MATH.CONTENT.HSF.IF.B.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.
• CCSS.MATH.CONTENT.HSF.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• CCSS.MATH.CONTENT.HSF.BF.A.1 — Write a function that describes a relationship between two quantities.
• CCSS.MATH.CONTENT.8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
• CCSS.MATH.CONTENT.HSF.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
• CCSS.MATH.CONTENT.7.SP.C.8.B — Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
• CCSS.MATH.CONTENT.HSF.BF.B.4.C — (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.