Functions

Extend the concept of a function by recognizing that trigonometric ratios are functions of angle measure.

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

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities.

Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.

Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Write a function that describes a relationship between two quantities.

Build polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

Build a new function, in terms of a context, by combining standard function types using arithmetic operations.

Extend an understanding of the effects on the graphical and tabular representations of a function when replacing 𝑓(𝑥) with 𝑘 ∙ 𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) to include 𝑓(𝑘 ∙ 𝑥) for specific values of 𝑘 (both positive and negative).

Find an inverse function.

Understand the inverse relationship between exponential and logarithmic, quadratic and square root, and linear to linear functions and use this relationship to solve problems using tables, graphs, and equations.

Determine if an inverse function exists by analyzing tables, graphs, and equations.

If an inverse function exists for a linear, quadratic and/or exponential function, f, represent the inverse function, f -1 , with a table, graph, or equation and use it to solve problems in terms of a context.

Compare the end behavior of functions using their rates of change over intervals of the same length to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a polynomial function.

Use logarithms to express the solution to 𝑎𝑏 𝑐𝑡 = 𝑑 where 𝑎, 𝑏, 𝑐, and 𝑑 are numbers and evaluate the logarithm using technology.

Understand radian measure of an angle as: • The ratio of the length of an arc on a circle subtended by the angle to its radius. • A dimensionless measure of length defined by the quotient of arc length and radius that is a real number. • The domain for trigonometric functions.

Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions.

Interpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate.

Interpret the cosine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.

Use technology to investigate the parameters, 𝑎, 𝑏, and ℎ of a sine function, 𝑓(𝑥) = 𝑎 ∙ 𝑠𝑖𝑛(𝑏 ∙ 𝑥) + ℎ, to represent periodic phenomena and interpret key features in terms of a context.