Section 3. Properties of a Function s Graph Objectives Find the intercepts of a function given its formula. Given the graph of a function, identify the domain and range of the function. Approximate relative minimum and maximum values and intercepts from a graph. Given the graph of a function, identify open intervals where the graph is increasing/decreasing/constant. Given the graph of a function, identify intervals where the graph is positive/negative. Given the equation or graph of a function, determine whether the function is odd, even, or neither. Given a partial graph or partial table of values for an even/odd function, complete the graph/table. Use a calculator to approximate features on a graph, such as zeros, maximums, and minimums. Use a calculator to answer questions about a function in a given context. Given certain characteristics of a function, such as the domain, range, and/or intervals on which it is increasing/decreasing or positive/negative, sketch a possible graph of the function. Sketch a complete graph of a function which includes all important features. Using your knowledge of domain and range, as well as context provided, find an appropriate viewing window for a function on a graphing calculator. Preliminaries What are intercepts and how are they found? What is meant by a zero of a function and are zeros found? Page 55
A function f(x) is increasing on an open interval if. A function f(x) is decreasing on an open interval if. A function f(x) is constant on an open interval if. What is a relative minimum? What is a relative maximum? What is an even function? What kind of symmetry does the graph of an even function have? What is an odd function? What kind of symmetry does the graph of an odd function have? Page 56
Warm-up. Solve for x. (A) 0 = x 3 x +x 8 (B) 0 = x+5 x 4 (C) 0 = x +5x+4 x 6. If f(x) = x 3x, evaluate f( x). Page 57
Class Notes and Examples How can you find the zeros of a function given by an equation? How could you check your solutions? 3.. Determine the zeros of the following functions algebraically. (A) y = 5x (B) p(n) = n n n n 6 Page 58
(C) w(x) = x 4 (D) z(x) = 3x + 5 x Page 59
How do you find the domain and range of a function, given the graph? 3.. Determine the domain and range of the function shown in the graph below: 5 4 3-5 -4 - - 3 4 5-4 -5-6 Page 60
How do identify the open interval(s) on which a function is increasing, decreasing or constant? 3..3 Identify the open intervals on which the function below is increasing, decreasing, and constant. 6 5 4 3-5 -4 - - 3 4 5 6 Page 6
How do identify the open interval(s) on which a function is positive or negative? 3..4 Identify the intervals on which the function below is positive and negative. Also list all zeros on the graph. 5 4 3-5 -4 - - 3 4 5-4 -5 Page 6
3..5 Use the graph of y = f(x) given below to answer the following questions. 5 4 3-5 -4-3 4 5 - -4 (A) What is f(3)? -5 (B) What is/are the x-intercept(s)? (C) Determine the domain and range. Page 63
3..6 Use the graph of y = f(x) given below to answer the following questions. 5 4 3-5 -4-3 4 5 - -4-5 (A) Determine the intervals on which f(x) is increasing, decreasing and constant. (B) Determine the relative maxima and minima. (C) For what value(s) is f(x) = 4? (D) For what value(s) is f(x) positive? negative? Page 64
3..7 Sketch the graph of a function that has the properties described. (A) A function whose range is (0, ) which is increasing on the interval ( 3, 5) and decreasing on the intervals (, 3) and (5, ). (B) A function whose domain is [ 4, 4) and range is [, ) that is decreasing on the interval ( 4, ) and increasing on the interval (, 4). Page 65
Given the graph of a function, how do you tell whether it is even, odd, or neither? 3..8 Determine if the functions graphed below are even, odd, or neither. (A) - 3 4 5 6 - -4 (B) 3-4 - 3 4 - (C) 6 5 4 3-4 - - 3 4-4 -5-6 Page 66
3..9 Complete the graph for negative values of x if the function is (A) Even (B) Odd 3..0 Complete the table if the function is (A) Even (B) Odd x 0 x 0 f(x) 5 3 f(x) 5 3 Given a function represented by an equation, how do you tell whether it is even, odd, or neither? Page 67
3.. Determine if the functions represented below are even, odd, or neither. (A) f(x) = 7x 3 + 5x + (B) g(x) = 4 x (C) h(x) = x x +0 Page 68