Standards - Mathematics

Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

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. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.

Find the difference quotient $\frac {f(x+\Delta x) - f(x)}{\Delta x}$ of a function and use it to evaluate the average rate of change at a point.

Find the difference quotient $\frac{f(x+\Delta x) - f(x)}{\Delta x}$ of a function and use it to evaluate the average rate of change at a point.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.

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

Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.

Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.

Given that a function has an inverse, write an expression for the inverse of the function.

Given that a function has an inverse, write an expression for the inverse of the function.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Produce an invertible function from a non-invertible function by restricting the domain.

Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.

Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f(k \cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.

Graph conic sections given their standard form.

Graph conic sections given their standard form.

Solve application-based problems involving parametric and polar equations.

Graph parametric and polar equations.

Convert parametric and polar equations to rectangular form.

Use special triangles to determine geometrically the values of sine, cosine, and tangent for $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$, and use the unit circle to express the values of sine, cosine, and tangent for $\pi - x$, $\pi + x$, and $2\pi - x$ in terms of their values for $x$, where $x$ is any real number.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Use trigonometric identities to solve problems.

Use the Pythagorean identity $sin^2 ( heta) + cos^2( heta) = 1$ to derive the other forms of the identity.

Use the Pythagorean identity $\sin^2 (\theta) + \cos^2(\theta) = 1$ to derive the other forms of the identity.

Use the Pythagorean and double angle identities to prove other simple identities.