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1.5 Analyzing Graphs of Functions (2,4) Find: a. the domain [-1,4) b. the range [-5,4] c. f(-1) = -5 (4,0) (-1,-5) d. f(2) = 4 Vertical Line Test for Functions Do the graphs represent y as a function of x? no yes yes Increasing and Decreasing Functions -2 -1 1 2 3 4 5 1. The function is decreasing on the interval (-2,0). 2. The function is constant on the interval (0,3). 3. The function is increasing on the interval (3,5). A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x 2 implies f(x1) < f(x2) A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x 2 implies f(x1) > f(x2) A function f is constant on an interval if, for any x1 and x2 in the interval, f(x1) = f(x2) go to page 57 Tests for Even and Odd Functions A function is y = f(x) is even if, for each x in the domain of f, f(-x) = f(x) An even function is symmetric about the y-axis. A function is y = f(x) is odd if, for each x in the domain of f, f(-x) = -f(x) An odd function is symmetric about the origin. Ex. g(x) = x3 - x g(-x) = (-x)3 – (-x) = -x3 + x = -(x3 – x) Therefore, g(x) is odd because f(-x) = -f(x) Ex. h(x) = x2 + 1 h(-x) = (-x)2 + 1 = x2 + 1 h(x) is even because f(-x) = f(x) Summary of Graphs of Common Functions f(x) = c y=x y x y x y=x3 y = x2 The Greatest Integer Function y [[ x]] x y 0 .2 .5 .8 .99 0 0 0 0 0 -4 -3 x y 1.1 1.4 1.7 1.8 1.99 1 1 1 1 1 -2 -1 0 1 2 1 3 2 4 5 3 Average Rate of Change of a Function Find the average rates of change of f(x) = x3 - 3x from x1 = -2 to x2 = 0. The average rate of change of f from x1 to x2 y f (x 2 ) f (x1) x x 2 x1 f (x 2 ) f (x1 ) f (0) f (2) 0 (2) 1 x 2 x1 0 2 2 Finding Average Speed The average speed of s(t) from t1 to t2 is s s(t 2 ) s(t1 ) t t 2 t1 Ex. The distance (in feet) a moving car is from a stoplight is given by the function s(t) = 20t3/2, where t is thetime (in seconds). Find the average speed of the car from t1 = 0 to t2 = 4 seconds. s(t 2 ) s(t1 ) s(4) s(0) 160 0 40 ft /sec t 2 t1 40 4 What’s the average speed of the car from 540160 76ft /sec 5 4 to 9 seconds?