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

Copyright © 2000 by the McGraw-Hill Companies, Inc. The Cartesian Coordinate System 1-1-1

Copyright © 2000 by the McGraw-Hill Companies, Inc. Distance Between Two Points | y 2 – y 1 | | x 2 – x 1 |

Copyright © 2000 by the McGraw-Hill Companies, Inc. Standard Equation of a Circle Circle with radius r and center at (h,k): ( x – h ) 2 + ( y – k ) 2 = r 2 r > 0 Circle P(x, y) C(h, k)

Copyright © 2000 by the McGraw-Hill Companies, Inc. An image on the screen of a graphing utility is made up of darkened rectangles called pixels. The pixel rectangles are the same size, and do not change in shape during any application. Graphing utilities use pixel-by-pixel plotting to produce graphs. Graphing Utility Screens Image Magnification to show pixels The portion of a rectangular coordinate system displayed on the graphing screen is called a viewing window and is determined by assigning values to six window variables: the lower limit, upper limit, and scale for the x axis; and the lower limit, upper limit, and scale for the y axis

Copyright © 2000 by the McGraw-Hill Companies, Inc. Definition of a Function 1-3-5

Copyright © 2000 by the McGraw-Hill Companies, Inc. Functions Defined by Equations In an equation in two variables, if to each value of the independent variable there corresponds exactly one value of the dependent variable, then the equation defines a function. If there is any value of the independent variable to which there corresponds more than one value of the dependent variable, then the equation does not define a function. The equation y = x 2 – 4 defines a function. The equation x 2 + y 2 = 16 does not define a function. The equation y = x 2 – 4 defines a function. The equation x 2 + y 2 = 16 does not define a function

Copyright © 2000 by the McGraw-Hill Companies, Inc. Vertical Line Test for a Function An equation defines a function if each vertical line in the rectangular coordinate system passes through at most one point on the graph of the equation. If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function. (A) 4y – 3x = 8(B) y 2 – x 2 =

Copyright © 2000 by the McGraw-Hill Companies, Inc. If a function is defined by an equation and the domain is not indicated, then we assume that the domain is the set of all real number replacements of the independent variable that produce real values for the dependent variable. The range is the set of all values of the dependent variable corresponding to these domain values. The symbol f(x) represents the real number in the range of the function f corresponding to the domain value x. Symbolically, f: x  f(x). The ordered pair (x, f(x)) belongs to the function f. If x is a real number that is not in the domain of f, then f is not defined at x and f(x) does not exist. Agreement on Domains and Ranges The Symbol f(x)

Copyright © 2000 by the McGraw-Hill Companies, Inc. x f(x) 5–5 10 –10 f(x) = –x 3 0 x g(x) 5–5 5 g(x) = 2x (a) Decreasing on (–   ) (b) Increasing on (– ,  ) Increasing, Decreasing, and Constant Functions 1-4-9(a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. x h(x) 5–5 5 h(x) = 2 0 x p(x) 5–5 5 p(x) = x 2 – 1 Increasing, Decreasing, and Constant Functions (c) Constant on (–   ) (d)Decreasing on (– , 0] Increasing on [0,  ) 1-4-9(b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. The functional value f(c) is called a local maximum if there is an interval (a, b) containing c such that f(x)  f(c) for all x in (a, b). The functional value f(c) is called a local minimum if there is an interval (a, b) containing c such that f(x)  f(c) for all x in (a, b). Local Maxima and Local Minima

Copyright © 2000 by the McGraw-Hill Companies, Inc. 5–5 5 x Identity Function f(x) f(x) = x 5–5 5 x Absolute Value Function g(x) = |x| g(x) Six Basic Functions (a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Six Basic Functions 5–5 5 Square Function x h(x) = x 2 h(x) 5–5 5 Cube Function x m(x) = x 3 m(x) (b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Six Basic Functions 5 5 Square-Root Function x n(x) =x n(x) 5–5 5 Cube-Root Function x p(x) = 3 x p(x) (c)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Graph Transformations Vertical Translation: y = f(x) + k Horizontal Translation: y = f(x+h) Reflection: y = – f(x)Reflect the graph of y = f(x) in the x axis Vertical Expansion and Contraction: y = A f(x) k > 0 Shift graph of y = f(x) up k units k < 0 Shift graph of y = f(x) down  k  units h > 0 Shift graph of y = f(x) left h units h < 0 Shift graph of y = f(x) right  h  units A > 1Vertically expand graph of y = f(x) by multiplying each ordinate value by A 0 < A < 1Vertically contract graph of y = f(x) by multiplying each ordinate value by A