Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen College Algebra: A Graphing Approach Chapter Two Linear and Quadratic Functions

Copyright © 2000 by the McGraw-Hill Companies, Inc. Linear and Constant Functions A function f is a linear function if f(x) = mx + b m  0 where m and b are real numbers. The domain is the set of all real numbers and the range is the set of all real numbers. If m = 0, then f is called a constant function f(x) = b which has the set of all real numbers as its domain and the constant b as its range

Copyright © 2000 by the McGraw-Hill Companies, Inc. Slope of a Line x 1  x (x 2, y 1 )

Copyright © 2000 by the McGraw-Hill Companies, Inc. Geometric Interpretation of Slope LineSlopeExample RisingPositive FallingNegative HorizontalZero VerticalNot defined

Copyright © 2000 by the McGraw-Hill Companies, Inc. Standard form Ax + By = CA and B not both 0 Slope-intercept form y = mx + b Slope: m ; y intercept: b Point-slope form y – y 1 = m ( x – x 1 )Slope: m ; Point: ( x 1, y 1 ) Horizontal line y = b Slope: 0 Vertical line x = a Slope: Undefined Equations of a Line

Copyright © 2000 by the McGraw-Hill Companies, Inc. An equivalent inequality will result and the sense will remain the same if each side of the original inequality: 1. Has the same real number added to or subtracted from it; or 2. Is multiplied or divided by the same positive number. An equivalent inequality will result and the sense will reverse if each side of the original inequality: 3.Is multiplied or divided by the same negative number. Note: Multiplication by 0 and division by 0 are not permitted. Inequality Properties

Copyright © 2000 by the McGraw-Hill Companies, Inc. Completing the Square To complete the square of the quadratic expression x 2 + bx add the square of one-half the coefficient of x; that is, add or The resulting expression can be factored as a perfect square: x 2 + bx + =

Copyright © 2000 by the McGraw-Hill Companies, Inc. Given a quadratic function f(x) = ax 2 + bx + c, a  0, and the form f(x) = a (x – h) 2 + k obtained by completing the square: 1.The graph of f is a parabola. 2.Vertex: (h, k) [parabola increases on one side of vertex and decreases on the other]. 3.Axis (of symmetry): x = h (parallel to y axis) 4.f(h) = k is the minimum if a > 0 and the maximum if a < 0 5.Domain: All real numbers Range: (– ,k] if a 0 6.The graph of f is the graph of g(x) = ax 2 translated horizontally h units and vertically k units. Properties of a Quadratic Function (h, k)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Basic Properties of the Complex Number System 1.Addition and multiplication of complex numbers are commutative and associative. 2.There is an additive identity and a multiplicative identity for complex numbers. 3.Every complex number has an additive inverse (that is, a negative). 4.Every nonzero complex number has a multiplicative inverse (that is, a reciprocal). 5.Multiplication distributes over addition. 1.Addition and multiplication of complex numbers are commutative and associative. 2.There is an additive identity and a multiplicative identity for complex numbers. 3.Every complex number has an additive inverse (that is, a negative). 4.Every nonzero complex number has a multiplicative inverse (that is, a reciprocal). 5.Multiplication distributes over addition

Copyright © 2000 by the McGraw-Hill Companies, Inc. The Quadratic Formula If ax 2 + bx + c = 0, a  0, then Discriminant b 2 – 4ac Roots of ax 2 + bx + c = 0 No. of real zeros of f(x) = ax 2 + bx + c Positive2 distinct real roots2 01 real root (double root)1 Negative2 imaginary roots, one the conjugate of the other Discriminants, Roots, and Zeros

Copyright © 2000 by the McGraw-Hill Companies, Inc. Power Operation on Equations If both sides of an equation are raised to the same natural number power, then the solution set of the original equation is a subset of the solution set of the new equation. Extraneous solutions may be introduced by raising both sides of an equation to the same power. Every solution of the new equation must be checked in the original equation to eliminate extraneous solutions. EquationSolution set x = 3{ 3 } x 2 = 9{ – 3, 3 }