Unit 2 Functions Analyzing Graphs of Functions (Unit 2.2)

Transcription

1 Unit 2 Functions Analzing Graphs of Functions (Unit 2.2) William (Bill) Finch Mathematics Department Denton High School Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Lesson Goals When ou have completed this lesson ou will: Find the domain and range of functions. Appl the vertical line test and find the zeros of a function. Use functions to model and solve real-world problems. Identif increasing, decreasing and constant intervals of functions. Find relative etrema of functions. Graphs 2 / 25

2 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Lesson Goals When ou have completed this lesson ou will: Find an average rate of change. Identif even and odd functions. Graphs 3 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Graph of a Function The graph of a function f is the collection of ordered pairs (, f ()) such that is in the domain of f. = the directed distance from the -ais = f () = f () = the directed distance from the -ais f () Graphs 4 / 25

3 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Domain and Range of a Function The domain of a function is an interval that describes, from left to right, all of the input values to the function. The range of a function is an interval that describes, from bottom to top, all of the output values from the function. Range Domain Graphs 5 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 1 Identif the domain and range of the function shown below Graphs 6 / 25

4 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Vertical Line Test for a Function If a vertical line can be drawn anwhere on the graph of a relation and have at most one point of intersection on the graph, then the relation is a function. Can ou eplain wh? This is the graph of as a function of. This is not the graph of as a function of. Graphs 7 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Zeros of a Function The zeros of a function f are -values for which f () = 0. (1, 0) (2, 0) Graphs 8 / 25

5 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 2 Find the zeros of the function f () = Graphs 9 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 3 Find the zeros of the function g() = 5 2. Graphs 10 / 25

6 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 4 Find the zeros of the function h() = Graphs 11 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Increasing and Decreasing Functions As ou read from left to right the graph of a function is considered to be: Decreasing 1 < 2 f ( 1 ) > f ( 2 ) Constant 1 < 2 f ( 1 ) = f ( 2 ) Increasing 1 < 2 f ( 1 ) < f ( 2 ) Decreasing Constant Increasing Graphs 12 / 25

7 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 5 Determine the intervals over which the function is: a) decreasing 8 6 b) constant 4 2 c) increasing Graphs 13 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 6 Determine the intervals over which the function is: a) decreasing 4 b) constant c) increasing Graphs 14 / 25

8 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Relative Minimum and Relative Maimum The points at which a function changes its decreasing, constant, or increasing behavior are called the relative minimum or relative maimum values of the function. Rel Ma Rel Min Graphs 15 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Relative Minimum and Relative Maimum A function value f (a) is called a relative minimum of f if there eists and interval ( 1, 2 ) that contains a such that: 1 < < 2 f (a) f () A function value f (a) is called a relative maimum of f if there eists an interval ( 1, 2 ) that contains a such that: 1 < < 2 f (a) f () Graphs 16 / 25

9 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 7 Finding relative minima and maima eactl is a job for Calculus. Meanwhile, we can use a graphing utilit (such as a graphing calculator) to provide approimate values. Find approimate values for the relative minima and maima of function f : f () = Graphs 17 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Average Rate of Change On a non-linear graph the rate of change (slope) changes at each point. The average rate of change between an two points is the slope of the line through those two points. (1, f (1)) f (2, f (2)) Avg Rate of Change = = f ( 2) f ( 1 ) 2 1 Secant line Graphs 18 / 25

10 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 8 For the function f () = find the average rate of change from 1 = 4 to 2 = 2. Graphs 19 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Even and Odd Functions The graph of an even function is smmetric wrt the -ais. f (, ) (, ) A function = f () is even if, for each in the domain of f, f ( ) = f (). Graphs 20 / 25

11 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Even and Odd Functions The graph of an odd function is smmetric wrt the origin. f (, ) (, ) A function = f () is odd if, for each in the domain of f, f ( ) = f (). Graphs 21 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 9 Show that f () = 3 + is an odd function. Graphs 22 / 25

12 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar Eample 10 Show that g() = 2 1 is an even function. Graphs 23 / 25 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar What You Learned You can now: Find the domain and range of functions. Appl the vertical line test and find the zeros of a function. Use functions to model and solve real-world problems. Identif increasing, decreasing and constant intervals of functions. Find relative etrema of functions. Graphs 24 / 25

13 Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even Summar What You Learned You can now: Find an average rate of change. Identif even and odd functions. Do problems Chap 1.2 #9-15 odd, odd, odd, 42, odd, 55; Chap 1.4 #1, 7, 23, 27, 35, 39, 43, 47, 51 Graphs 25 / 25