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 (4, 6)6f (4) = (4) 2 – 2(4) – 2 = 64 (3, 1)1f (3) = (3) 2 – 2(3) – 2 = 13 -2 -3 -2 1 6 y (2, -2)f (2) = (2) 2 – 2(2) – 2 = -22 (1, -3)f (1) = (1) 2 – 2(1) – 2 = -31 (0, -2)f (0) = (0) 2 – 2(-0) – 2 = -20 (-1, 1)f (-1) = (-1) 2 – 2(-1) – 2 = 1 (-2, 6)f (-2) = (-2) 2 – 2(-2) – 2 = 6-2 (x, y)f (x) = x 2 – 2x – 2x x y (4, 6)(-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

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 The zeros of a function y = f (x) are the values of x for which y = 0. Example: The zeros of f (x) = 4 + 3x – x 2 can be found algebraically by setting f (x) = 0 and solving for x. 0 = 4 + 3x – x 2 Set f (x) = 0. (4 – x)(1 + x) Factor. x = -1, 4 Solve for x. The zeros of f (x) = 4 + 3x – x 2 are x = -1 and x = 4. The zeros of f (x) = 4 + 3x – x 2 can also be found geometrically by observing where the graph intersects the x-axis. x y 2 2 y = 4 + 3x – x 2 (4, 0) (-1, 0) Zeros of a Function

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 A function is: ● increasing on an interval if, for every x 1 and x 2 in the interval, if x 1 < x 2, then f (x 1 ) < f (x 2 ) ● decreasing on an interval if, for every x 1 and x 2 in the interval, if x 1 f (x 2 ) ● constant on an interval if, for every x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). The graph of y = f (x): ● Increases on [-∞, -3] ● Decreases on [-3, 3] ● Increases on [3, + ∞]. (3, -4) x y (-3, 6) Increasing, Decreasing, and Constant Functions

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 A function is even if f (-x) = f (x) for every x in the domain of f. A function is odd if f (-x) = -f (x) for every x in the domain of f. A function f (x) = x 2 is even function, since f (-x) = (-x) 2 = x 2 = f (x). A function f (x) = x 3 is an odd function, since f (-x) = (-x) 3 = -x 3 = -f (x). Even and Odd Functions

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 A function is even if and only if its graph has line symmetry in the y-axis. The graphs shown have line symmetry in the y-axis. These functions are even. x y f (x) = x 2 x y f (x) = |x| x y f (x) = cos(x) Example: Even Function

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 A function is odd if and only if its graph has point symmetry through the origin. The graphs shown have point symmetry through the origin. These functions are odd. x y f (x) = x 3 f (x) = sin(x) x y f (x) = x y Example: Odd Function