Name: Period: Date: Analyzing Graphs of Functions and Relations Guided Notes

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1 Analzing Graphs of Functions and Relations Guided Notes The graph of a function f is the set of ordered pairs(, f ), in the coordinate plane, such that is the domain of f. the directed distance from the -ais = f the directed distance from the -ais You can use the graph to estimate function values. Sample Problem : Use a graph of each function to estimate the indicated function values. Then find the values algebraicall. a. f = f =? f =? Graphicall f = f = b. f = + + f 0 =? f =? f =? Algebraicall f = = 0 = f = = = = Graphicall f 0 = f = 0 f = Algebraicall f 0 = = f = + + = 8 + = 0 f = + + = = Copright PreCalculusCoach.com

2 Analzing Graphs of Functions and Relations Guided Notes Identifing Intercepts from a Functions Graph A point where the graph intersects or meets the or ais is called an intercept. An -intercept occurs where = 0. A -intercept occurs where = 0. Sample Problem : Use the graph of each function to approimate its intercept. Then find the intercept algebraicall. a. g = b. f = Graphicall g = intercept = Graphicall f = + + intercept = Algebraicall -intercept occurs where = 0. g 0 = 0 = g 0 = intercept = Algebraicall -intercept occurs where = 0. f 0 = f 0 = intercept = Zeros of a Function The zeros of function f are values for which f = 0 If the graph of a function of has an -intercept at (, 0) then is a zero of the function. To find the zeros of a function, set the function equal to zero and solve for the independent variable. Copright PreCalculusCoach.com

3 Analzing Graphs of Functions and Relations Guided Notes Sample Problem : Use the graph of each function to approimate its zeros. Then find the zeros of each function algebraicall. a. f = Zeros =? f = b. f = + Zeros =? f = Graphicall Graphicall f = intercepts Algebraicall and 0 f = + intercepts. Algebraicall f = 0 = 0 + = 0 = 0 or + = 0 = The zeros of f are 0 and f = 0 + = 0 = = The zero of f is. Smmetr of Graphs There are two possible tpes of smmetr that graphs of functions can have.. Line smmetr - graphs can be folded along a line so that the two halves match eactl.. Point smmetr - graphs can be rotated 80 with respect to a point and appear unchanged. Copright PreCalculusCoach.com

4 Analzing Graphs of Functions and Relations Guided Notes Tests for Smmetr Graphical Test The graph of a relation is smmetric with respect to the -ais if and onl if for ever point (, ), on the graph, the point (, ), is also on the graph. The graph of a relation is smmetric with respect to the -ais if and onl if for ever point (, )on the graph, the point (, )is also on the graph. The graph of a relation is smmetric with respect to the origin if and onl if for ever point (, ) on the graph, the point (, )is also on the graph. Algebraic Test Replacing with - produces an equivalent equation. Replacing with - produces an equivalent equation. Replacing with - and with - produces an equivalent equation. Sample Problem : Use the graph of each equation to test for smmetr with respect to the -ais, -ais, and the origin. Support the answer numericall. Then confirm algebraicall. a. = Graphicall The graph appears to be smmetric with respect to the origin because for ever point (, ) on the graph, there is a point (, ) Support Numericall There is a table of values to support this conjecture. - - (, ) (, ) (, ) (, ) (, ) (, ) (, ) Algebraicall = Because = is equivalent to =, the graph is smmetric R with respect to the origin. Copright PreCalculusCoach.com

5 Analzing Graphs of Functions and Relations Guided Notes b. + = Graphicall The graph appears to be smmetric with respect to the -ais because for ever point (, ) on the graph, there is a point (, ) Support Numericall There is a table of values to support this conjecture. 0 ± ± ± 0 (, ), ± (, ± ) (, ±) (, 0) Algebraicall + = + = Because + = is equivalent to + =, the graph is smmetric with respect to the -ais. Identif Even and Odd Functions If f = f, then the function is even, and smmetric to the -ais. If f = f, then the function is odd, and smmetric to the origin. Sample Problem : Determine whether the following are even, odd, or neither. a. f = + f = + f = + f = + f = f The function is even. Copright PreCalculusCoach.com

6 Analzing Graphs of Functions and Relations Guided Notes b. g = 9 g = 9 g = 9 g = 9 + g = (9 ) g = g The function is odd. c. h t = t + t h t = t + t h t = t + ( t) h t = t t h t h t h t h t The function is neither. Copright PreCalculusCoach.com 6