1. Graphs of Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 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. 3. Connect them with a curve. To graph a function: 1. Make a table of values. 2. Plot the points. Steps to Graph a Function

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 xf (x) = x 2 – 2x – 2y(x, y) -2f (-2) = (-2) 2 – 2(-2) – 2 = 66(-2, 6) f (-1) = (-1) 2 – 2(-1) – 2 = 11(-1, 1) 0f (0) = (0) 2 – 2(-0) – 2 = -2-2(0, -2) 1f (1) = (1) 2 – 2(1) – 2 = -3-3(1, -3) 2f (2) = (2) 2 – 2(2) – 2 = -2-2(2, -2) 3f (3) = (3) 2 – 2(3) – 2 = 11(3, 1) 4f (4) = (4) 2 – 2(4) – 2 = 66(4, 6) x y (-2, 6) (0, -2) (-1, 1) (3, 1) (1, -3) (2, -2) Example: Graph the function f (x) = x 2 – 2x – 2. Example: Graph a Function

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 x y 4 -4 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. Example: Find the domain and range of f (x) = x 2 – 2x – 2 by investigating its graph. Domain = all real numbers Range The domain is the real numbers. The range is { y: y ≥ -3} or [-3, + ]. Domain & Range

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 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). The domain is [-3,+∞] or {x: x ≥ -3}. The range is [0,+∞] or {y: y ≥ 0}. Range Domain Example: Domain & Range x y (-3, 0)

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 x 2 + y 2 = 4 A relation is a correspondence that associates values of x with values of y. Example: The following equations define relations: The graph of a relation is the set of ordered pairs (x, y) for which the relation holds. y 2 = x x y (4, 2) (4, -2) x y (0, 2) (0, -2) y = x 2 x y Relation

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 y 2 = x x y 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 = x 2 is a function. Consider the graphs. x 2 + y 2 = 1 x y y = x 2 x y 2 points of intersection 1 point of intersection 2 points of intersection Vertical Line Test

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Vertical Line Test: Apply the vertical line test to determine which of the relations are functions. 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. x y x = 2y – 1 x y x = | y – 2| Example: Vertical Line Test