Lesson Goals. Unit 2 Functions Analyzing Graphs of Functions (Unit 2.2) Graph of a Function. Lesson Goals

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1 Unit Functions Analzing Graphs of Functions (Unit.) William (Bill) Finch Mathematics Department Denton High School 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 / 5 Lesson Goals When ou have completed this lesson ou will: Find an average rate of change. Identif even and odd functions. 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 3 / 5 Graphs / 5

2 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 Eample 1 Identif the domain and range of the function shown below Domain Graphs 5 / 5 Graphs 6 / 5 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? Zeros of a Function The zeros of a function f are -values for which f () = 0. (1, 0) (, 0) This is the graph of as a function of. This is not the graph of as a function of. Graphs 7 / 5 Graphs / 5

3 Eample Find the zeros of the function f () = Eample 3 Find the zeros of the function g() = 5. Graphs 9 / 5 Graphs 10 / 5 Eample Find the zeros of the function h() = Increasing and Decreasing Functions As ou read from left to right the graph of a function is considered to be: Decreasing 1 < f (1) > f () Constant 1 < f (1) = f () Increasing 1 < f (1) < f () Decreasing Constant Increasing Graphs 11 / 5 Graphs 1 / 5

4 Eample 5 Determine the intervals over which the function is: a) decreasing 6 b) constant c) increasing Eample 6 Determine the intervals over which the function is: a) decreasing b) constant c) increasing Graphs 13 / 5 Graphs 1 / 5 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. Relative Minimum and Relative Maimum A function value f (a) is called a relative minimum of f if there eists and interval (1, ) that contains a such that: 1 < < f (a) f () Rel Ma A function value f (a) is called a relative maimum of f if there eists an interval (1, ) that contains a such that: 1 < < f (a) f () Rel Min Graphs 15 / 5 Graphs 16 / 5

5 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 () = 3 3 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 (, f ()) Avg Rate of Change = f () f (1) = 1 Secant line Graphs 17 / 5 Graphs 1 / 5 Eample For the function f () = + 15 find the average rate of change from 1 = to =. 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 19 / 5 Graphs 0 / 5

6 Even and Odd Functions The graph of an odd function is smmetric wrt the origin. Eample 9 Show that f () = 3 + is an odd function. f (, ) (, ) A function = f () is odd if, for each in the domain of f, f ( ) = f (). Graphs 1 / 5 Graphs / 5 Eample 10 Show that g() = 1 is an even function. 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 3 / 5 Graphs / 5

7 What You Learned You can now: Find an average rate of change. Identif even and odd functions. Do problems Chap 1. #9-15 odd, 17-1 odd, odd,, 9-53 odd, 55; Chap 1. #1, 7, 3, 7, 35, 39, 3, 7, 51 Graphs 5 / 5