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1 Section. Analyzing Graphs of Functions , and,. m 6 y y Slope m y y y y y. 6, and, 6. m 6 9 y 6 9 9y 6 9y Slope 6 9 m 9 y 9 y 9 8 8y 8y 9 Section. Analyzing Graphs of Functions You should be able to determine the domain and range of a function from its graph. You should be able to use the vertical line test for functions. You should be able to find the zeros of a function. You should be able to determine when a function is constant, increasing, or decreasing. You should be able to approimate relative minimums and relative maimums from the graph of a function. You should know that f is odd if f f. even if f f.
2 6 Chapter Functions and Their Graphs Vocabulary Check. ordered pairs. vertical line test. zeros. decreasing. maimum 6. average rate of change; 7. odd 8. even secant. Domain:,, Range:,. Domain:,. Domain:, Range:, Range:,. Domain:,,,. f f 6. f f Range:, f (d) f f (d) f 7. f f 8. f f 9. y f (d) f f (d) f A vertical line intersects the graph just once, so y is a function of.. y. A vertical line intersects the graph no more than once, so y is a function of. y y ± y is not a function of. Some vertical lines cross the graph twice.. y. A vertical line intersects the graph more than once, so y is not a function of. y. A vertical line intersects the graph just once, so y is a function of. y A vertical line intersects the graph more than once, so y is not a function of or 6 or 6 f f or f ± 9 ± f ,, 9 ± 6,,
3 Section. Analyzing Graphs of Functions 7.. f Zero: Zeros:, 7 f f 8 6 Zero: 8 6 Zero: Zeros: ±. Zero: f ± ±.. f. f is increasing on,. f. The graph is decreasing on, and increasing on,. f f is increasing on, and,. f is decreasing on,.. f. The graph is decreasing on, and increasing on,., f,, f is increasing on, and,. f is constant on,. < > 6. f,, > The graph is decreasing on, and increasing on, and,.
4 8 Chapter Functions and Their Graphs 7. f 8. The graph is decreasing on and and, increasing on, and, f is increasing on,.. f is constant on,. f is decreasing on,., 9. f Constant on,. g Increasing on, f g. gs s 7. h Decreasing on, ; Increasing on, s g s Decreasing on, ; Increasing on, h. f t t. f 6 Increasing on, ; Decreasing on, t f t 6 6 Increasing on,,, ; Decreasing on,,, f. f Decreasing on, f
5 Section. Analyzing Graphs of Functions 9 6. f 7. f 9 Increasing on, Increasing on, ; Decreasing on, f.8. 8 f. 8. f 6 f.9.9 Decreasing on, ; Increasing on, 9. f. 8 f 7 8 Relative minimum:, 9 7 Relative minimum:, 6 or.,.. f. f 9. f 6 6 Relative maimum:.,. Relative maimum:.,. Relative minimum:.,.6 Relative maimum:.79, 8.. f f f on,. y Relative maimum:.,.8 Relative minimum:.,.8
6 Chapter Functions and Their Graphs 6. f y 7. f, f f on, and,. y 8. f y 9. f,,, f f on,. y 6. f y 6. f, f f is never greater than. ( f < for all.) y 6. f y 6. f is always greater than., f f f 9 The average rate of change from to is. 6. f 8 6. f f The average rate of change from to is. f f f The average rate of change from to is f f f f 7 6 The average rate of change from to is. f f The average rate of change from to is.
7 Section. Analyzing Graphs of Functions 68. f f 6 f The average rate of change from to 6 is. f f f 8 The average rate of change from to is. 7. f 7. f 8 f 8 The average rate of change from to is 8. f 6 f 6 6 f The function is even. y-ais symmetry 7. h 7. h h h The function is neither odd nor even. No symmetry g 7. g g The function is odd. Origin symmetry f f f The function is odd. Origin symmetry 7. f t t t 76. ft t t t t ft, ft The function is neither even nor odd. No symmetry gs s 77. gs s s gs The function is even. y-ais symmetry h top bottom 78. h top bottom 79. h top bottom 8. h top bottom 8. L right left 8. L right left 8. L right left y y y y 8. L right left 8. L , 9 y 6 y 9 L when 9.96 watts.
8 Chapter Functions and Their Graphs The model is an ecellent fit. The temperature is increasing from 6 A.M. until noon to 6. Then it decreases until A.M. 6 to. Then the temperature increases until 6 A.M. to. (d) The maimum temperature according to the model is about 6.9F. According to the data, it is 6F. The minimum temperature according to the model is about.98f. According to the data, it is F. (e) Answers may vary. Temperatures will depend upon the weather patterns, which usually change from day to day. 87. For the average salaries of college professors, a scale of $, would be appropriate. For the population of the United States, use a scale of,,. For the percent of the civilian workforce that is unemployed, use a scale of % m 8 When, the resulting figure is a square. 8 m 8 m Range: A 6 8 m s A 88 6 Domain: By the Pythagorean Theorem, s s meters. 89. r.69t.7t.t, t 7 7 r7 r The average rate of change from to 7 is $8.6 billion per year. The estimated revenue is increasing each year at a rapid pace The average rate of change from 99 to : F F The number of foreign students increased at a steady rate of.78 thousand students each year. The five-year period of least average rate of change was 99 to 997. F 7 F 7 The five-year period of greatest increase was 997 to. F F The least rate of change was about 6. thousand students from 99 to 997. The greatest rate of change was about. thousand students from 997 to.
9 Section. Analyzing Graphs of Functions 9. s 6, v 6 9. s 6t 7t 6. s 6t 6t 6 (d) The average rate of change of the height of the object with respect to time over the interval t to t is 6 feet per second. (e) (f) s s s 6, m 6 Secant line: 6 6 y 6 6t y 6t 6 The average rate of change from t to t : s s (d) The slope of the secant line through, s and, s is positive. The average rate of change of the position of the object from t to t is 8 feet per second. (e) The equation of the secant line: m 8, y 8t 6. (f) The graph is shown in feet per second 9. v, s s 6t t 7 9. s 6t 96t 7 6 (d) The average decrease in the height of the object over the interval t to t is 8 feet per second. (e) (f) s s s, m 8 Secant line: y 8t y 8t 8 8 The average rate of change from t to t : s s (d) The slope of the secant line through, s and, s is negative. The average rate of change of the position of the object from t to t is 6 feet per second. (e) The equation of the secant line: Using, s, 8 we have y 8 6t y 6t 6. (f) The graph is shown in. 6 feet per second m 6 7 6
10 Chapter Functions and Their Graphs 9. v, s 96. s 6t 8 s 6t (d) On the interval t to t, the height of the object is decreasing at a rate of feet per second. (e) (f) s s s, m Secant line: 6 y t y t The average rate of change from t to t : s s (d) The slope of the secant line through, s and, s is negative. The average rate of change of the position of the object from t to t is 8 feet per second. (e) The equation of the tangent line: Using, s, 6 we have y 6 8t (f) The graph is shown in. y 8t. 8 feet per second m False. The function f has a domain of all real numbers. 98. False. An odd function is symmetric with respect to the origin, so its domain must include negative values. 99. Even. The graph is a reflection in the -ais. Even. The graph is a reflection in the y-ais. Even. The graph is a vertical translation of f. (d) Neither. The graph is a horizontal translation of f.. Yes, the graph of y in Eercise does represent as a function of y. Each y-value corresponds to only one -value..,. If f is even, another point is,. If f is odd, another point is,., 7 If f is even, another point is, 7. If f is odd, another point is, 7.., 9., If f is even, another point is, 9. If f is even, another point is,. If f is odd, another point is, 9. If f is odd, another point is,.
11 Section. Analyzing Graphs of Functions. y y y (d) y (e) y (f) y 6 All the graphs pass through the origin. The graphs of the odd powers of are symmetric with respect to the origin and the graphs of the even powers are symmetric with respect to the y-ais. As the powers increase, the graphs become flatter in the interval < <. 6. The graph of y 7 will pass through the origin and will be symmetric with the origin. The graph of y 8 will pass through the origin and will be symmetric with respect to the y-ais or ± or ±. 6. f 8 f f 8 8 f f f 6 f f
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