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Course 2, Unit 5 - Nonlinear Functions and Equations

This is the second unit whose primary focus is quadratic functions and equations. The Course 1 unit Quadratic Functions was intended to help students begin to develop a clear and connected understanding of the numeric, graphic, verbal, and symbolic representations of quadratic functions and the ways that those representations can be applied to quantitative problems and patterns in data. Students should bring an intuitive understanding of quadratic patterns of change and some technical skills for reasoning with the various representations of those patterns to this unit.

Students will review and practice previously learned concepts and skills related to quadratic functions and equations, as well as expand their symbol manipulation skills to more sophisticated problems both in and out of context, through two lessons in this unit.

Further work toward developing proficiency with manipulating symbols and transforming function representation occurs in subsequent units and courses. Additional practice for the skills developed in this unit is incorporated in Review tasks of the On Your Own homework sets of later units.

Key Ideas from Course 2, Unit 5

  • Functions and function notation: We say that a relationship between two variables is a function when each value of the independent variable x corresponds to exactly one value of the dependent variable y. An example of a function and a nonfunction are shown below with a few examples of the use of function notation. Not all functions, such as the one below, can be defined by algebraic rules. (See pp. 327-329.)

    Function Non-function
  • Domain and range: For a function f(x), it is customary to refer to the variable x as the input and the function value f(x) as the output. The input values are referred to as the domain of the function. The output values are referred to as the range of the function. (See p. 330.)

  • Constructing quadratic rules: Given key points on a graph of a quadratic function, a rule can be found so that the graph of the rule contains the points. (See p. 333.)

  • Expanding and factoring quadratic expressions: The Distributive Property can be applied to expand and factor quadratic expressions. (See p. 337 Problem 2 and p. 338 Problem 8.)

  • Quadratic formula: The quadratic formula was initially developed in Course 1, Unit 7. (The formula and an explanation of its use are on p. 342 of the Course 2 student text.)

  • Solving quadratic equations: Factoring techniques and the quadratic formula can be used to solve quadratic equations. (See pp. 340-344.)

  • Solving equations and inequalities: Numeric, graphic, and symbolic strategies can be used to solve equations and inequalities involving comparisons between a linear and an inverse variation function or a linear and a quadratic function. The solution to the equation

    x2 - 5x -4 = -7x + 4
    x = -4 and x = 2

    is shown in the graph below. (See pp. 360-367.) Solving these types of systems draws on techniques for solving quadratic equations.

  • Common logarithms (base 10): The concept of the base 10 logarithm is developed from the problem of solving equations such as 10x = 9.5. The definition of a common logarithm is usually expressed in function-like notation: log10 a = b if and only if 10b = a. So, log10 9.5 = x. (See p. 370 Problems 1-3. Only common logarithms are developed here. Logarithms with other bases and the properties of logarithms are developed in Core-Plus Mathematics Courses 3 and 4.)

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