Standards / Objectives
- Solve quadratic equations with real coefficients that have complex solutions
- Use the structure of an expression to identify ways to rewrite it.
- Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
- . Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
- . Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- 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 hasthe larger maximum.
- . 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.
- . Define appropriate quantitiesfor the purpose of descriptive modelin
- Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ c. Use the properties of exponents to transform expressions for exponential functions.
- . Prove polynomial identities and use them to describe numerical relationships.
- 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 featuresinclude: intercepts; intervals where the function isincreasing, decreasing, positive, or negative; relative maximums and minimums;symmetries; end behavior; and periodicity. ★
- 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. ★
- . Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- 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 functionsfrom their graphs and algebraic expressionsfor them.
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LINKS
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Finding Zeros
Complex Roots
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