Digital Lesson Graphs of Functions

Steps to Graph a Function The graph of a function y = f (x) is a set of ordered pairs (x, f (x)), for values of x in the domain of f. To graph a function: 1. Make a table of values. 2. Plot the points. 3. Connect them with a curve. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Steps to Graph a Function

Example: Graph a Function Example: Graph the function f (x) = x2 – 2x – 2. x y x f (x) = x2 – 2x – 2 y (x, y) -2 f (-2) = (-2)2 – 2(-2) – 2 = 6 6 (-2, 6) -1 f (-1) = (-1)2 – 2(-1) – 2 = 1 1 (-1, 1) f (0) = (0)2 – 2(-0) – 2 = -2 (0, -2) f (1) = (1)2 – 2(1) – 2 = -3 -3 (1, -3) 2 f (2) = (2)2 – 2(2) – 2 = -2 (2, -2) 3 f (3) = (3)2 – 2(3) – 2 = 1 (3, 1) 4 f (4) = (4)2 – 2(4) – 2 = 6 (4, 6) (4, 6) (-2, 6) (0, -2) (-1, 1) (3, 1) (1, -3) (2, -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph a Function

The domain is the real numbers. The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. x y 4 -4 Example: Find the domain and range of f (x) = x2 – 2x – 2 by investigating its graph. Range The domain is the real numbers. The range is { y: y ≥ -3} or [-3, + ]. Domain = all real numbers Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Domain & Range

Example: Domain & Range Example: Find the domain and range of the function f (x) = from its graph. The graph is the upper branch of a parabola with vertex at (-3, 0). x y Range (-3, 0) Domain The domain is [-3,+∞] or {x: x ≥ -3}. The range is [0,+∞] or {y: y ≥ 0}. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Domain & Range

Example: The following equations define relations: y 2 = x y = x 2 A relation is a correspondence that associates values of x with values of y. The graph of a relation is the set of ordered pairs (x, y) for which the relation holds. Example: The following equations define relations: y 2 = x y = x 2 x 2 + y 2 = 4 x y (4, 2) (4, -2) x y x y (0, 2) (0, -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Relation

Vertical Line Test A relation is a function if no vertical line intersects its graph in more than one point. Of the relations y 2 = x, y = x 2, and x 2 + y 2 = 1 only y = x2 is a function. Consider the graphs. y 2 = x x y y = x 2 x y x 2 + y 2 = 1 x y 2 points of intersection 1 point of intersection 2 points of intersection Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Vertical Line Test

Example: Vertical Line Test Vertical Line Test: Apply the vertical line test to determine which of the relations are functions. y x = | y – 2| y x = 2y – 1 x x The graph does not pass the vertical line test. It is not a function. The graph passes the vertical line test. It is a function. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Vertical Line Test