This is a remix of a task to help students think about an expression for a function as built up out of simple operations on the variable, and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).

Students match features of exponential and logarithmic functions and then consider which features could change with a transformation. Features include: domain, range, intervals of increasing/decreasing, x- and y- intercepts, end behavior, and asymptotes.

Finding extrema and intervals increasing and decreasing for a function Mathematics Instructional Plans (MIPs) help teachers align instruction with the Mathematics Standards of Learning (SOL) by providing examples of how the knowledge, skills and processes found in the SOL and curriculum framework can be presented to students in the classroom.

In this module, students draw on their foundation of the analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. Students explore the role of factoring, as both an aid to the algebra and to the graphing of polynomials. Students continue to build upon the reasoning process of solving equations as they solve polynomial, rational, and radical equations, as well as linear and non-linear systems of equations. The module culminates with the fundamental theorem of algebra as the ultimate result in factoring. Students pursue connections to applications in prime numbers in encryption theory, Pythagorean triples, and modeling problems.
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In this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line and then extend their work with these functions to include solving exponential equations with logarithms. They use appropriate tools to explore the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations of a graph of a logarithmic function relate to the logarithmic properties. Students identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as "the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions" is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle, students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
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Students encountered the fundamental theorem of algebra, that every polynomial function has at least one zero in the realm of the complex numbers, in Algebra II Module 1. Topic A of this module brings students back to the study of complex roots of polynomial functions. Students first briefly review quadratic and cubic functions and then extend familiar polynomial identities to both complex numbers and to general polynomial functions. Students use polynomial identities to find square roots of complex numbers. The binomial theorem and its relationship to Pascal's triangle are explored using roots of unity. Topic A concludes with student's use of Cavalieri's principle to derive formulas for the volume of the sphere and other geometric solids.
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Students will use a draggable point to investigate intervals of increase and decrease and then practice writing and fining intervals.