- As an introduction to polynomial functions, students should be able to use quadratic functions in standard, factored, and vertex forms to graph and identify key features in context in order to answer questions about real-life phenomena.
- Key features of quadratic functions should include x and y-intercepts, roots, zeros, and solutions; domain, range, and intervals where the function is increasing, decreasing, positive, and/or negative (using inequality and interval notations); vertex, extreme value, and axis of symmetry; end behavior, using technology where appropriate.
- Students should be able to calculate the slope of average rate of change for a given interval, including the estimated rate of change.
- Through contextual exploration, students should recognize that there are data sets for which a quadratic function is not the best model, and therefore, explore other types of polynomial regression.
- Analysis of data sets with regressions should be done informally with verbal descriptions and with the use of technology.
- Students should be able to identify the real part of a complex number and the imaginary part.
- Students should convert any power of the imaginary unit, i, to an equivalent form and identify the pattern that emerges.
- Students should have opportunities to identify the complex conjugate of any complex number and recognize that complex numbers always occur as pairs when they represent solutions to a polynomial function.
- Students should be provided opportunities to solve real-life problems that require the addition, subtraction, or multiplication of complex numbers.
- Division of complex numbers is beyond the scope of Advanced Algebra.
- Expressions should include special-case quadratics such as perfect-square trinomials and the difference of two perfect squares.
- Equations and inequalities presented in real-life, mathematical problems should include quadratics with complex solutions.
- In previous grades, students had opportunities to create and solve quadratic equations. Given a real-life scenario, students should be able to model the scenario using quadratic equations and inequalities in one variable.
- Given a quadratic equation or inequality in one variable (model), students should be able to create a real-life scenario that matches the model.
- Students should be able to connect the solutions of quadratic equations and inequalities to the graph of the corresponding quadratic function and use these to solve real-life problems that can be modeled by quadratic equations and inequalities.
- Students should be able to model real-life occurrences with quadratic equations or inequalities and use these to solve problems.
- Students should be able to solve quadratic equations and inequalities fluently (flexibly, accurately, efficiently) by inspection, taking square roots, factoring, completing the square, and applying the quadratic formula, as appropriate to the initial form of the equation.
- Students should be provided opportunities to explore a variety of real-life problems modeled by quadratic equations and inequalities.
- Students can create equations involving areas using unknown dimensions.
- Students should be able to solve real-life problems modeled by systems of quadratic and linear functions using algebraic techniques (by hand) or using technology to identify the intersections of a parabola and a line.
- Students should be able to solve real-life problems modeled by quadratic equations and inequalities in two or more variables.
- Given a real-life scenario, students should be able to model the scenario using quadratic equations and inequalities in two or more variables.
- Given a quadratic equation or inequality in two or more variables (model), students should be able to create a real-life scenario that matches the model.
- Students can create equations involving projectile motion.
- Given a polynomial function, students should be able to apply the Fundamental Theorem of Algebra to describe the maximum number of times the function may cross the x-axis.
- Given a polynomial function, students should be able to tell if the left and right sides are increasing as x approaches negative and positive infinity based upon the sign of the leading coefficient and whether the greatest exponent is even or odd.
- Students should verify, using technology, if their predictions are correct for the number of zeros a polynomial has. When there are fewer zeros than the highest exponent led them to expect, students should understand that this is because there are complex solutions.
- Students should understand that complex solutions always occur in pairs.
- Students should be able to graph and identify key features of polynomial functions to include x and y-intercepts, roots of multiplicity, zeros, and solutions; domain, range, and intervals where the function is increasing, decreasing, positive, and/or negative (using inequality and interval notations); vertex, extreme value, and axis of symmetry; end behavior, using technology where appropriate.
- Students are not expected to graph polynomial functions presented in standard form, by hand.
- Students should be able to identify key features of a polynomial equation to create a rough sketch of a graph, by hand.
- When presented a polynomial function in standard form, students should be able to use technology to graph the function and identify key features, including the zeros of the function.
- Students should be able to rewrite polynomial expressions in various equivalent forms, based on the context of the problem.
- Students should be able to analyze a graph of a polynomial to identify where multiplicity exists due to a local maximum or minimum value that is situated on the x-axis and recognize that repeating that factor may be necessary.
- Student should note that multiplying factors to write a polynomial generates one of many possibly polynomials with the same zeros, because the leading coefficient of a polynomial is not evident from factor alone.
Textbook Connections
Module 5:
Lesson 1- Solving Polynomial Equations by Graphing
Lesson 2- Solving Polynomial Equations Algebraically
Lesson 5- Roots and Zeros
Module 3:
Lesson 3- Complex Numbers
Lesson 4- Factoring Quadratics
Lesson 5- Completing the Square
Lesson 6- Quadratic Formula and Discriminant
Lesson 7- Quadratic Inequalities
Lesson 8- Quadratic and Linear Systems of Equations
Module 5:
Lesson 1- Solving Polynomial Equations by Graphing
Lesson 2- Solving Polynomial Equations Algebraically
Lesson 5- Roots and Zeros
Module 3:
Lesson 3- Complex Numbers
Lesson 4- Factoring Quadratics
Lesson 5- Completing the Square
Lesson 6- Quadratic Formula and Discriminant
Lesson 7- Quadratic Inequalities
Lesson 8- Quadratic and Linear Systems of Equations