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 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

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 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

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Zero of a Function

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Finding the Zero of a Function

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Increasing and Decreasing Functions

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14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Average Rate of Change

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Even and Odd Functions a function is said to be even if its graph is symmetric with respect to the y-axis and to be odd if its graph is symmetric with respect to the origin.