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Chapter 5

Chapter 5. Functions and their Graphs. Function Notation. f(t) = h. Independent Variable Dependent Variable Example h = f(t) = 1454 – 16t 2. When t= 1, h= f(1)= 1438, We read as “f of 1 equals 1438” When t = 2, h = f(2) =1390, We read as ” f of 1 equals 1390 “.

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Chapter 5

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  1. Chapter 5 Functions and their Graphs

  2. Function Notation f(t) = h Independent Variable Dependent Variable Example h = f(t) = 1454 –16t2 When t= 1, h= f(1)= 1438, We read as “f of 1 equals 1438” When t = 2, h = f(2) =1390, We read as” f of 1 equals 1390 “

  3. Ch 5.1 (pg 251) Definition and Notation Table Ordered Pair Example –To rent a plane flying lessons cost $ 800 plus $30 per hour Suppose C = 30 t + 800 (t > 0) When t = 0, C = 30(0) + 800= 800 When t = 4, C = 30(4) + 800 = 920 When t = 10, C = 30(10) + 800 = 1100 The variable t in Equation is called the independent variable, and C is the dependent variable, because its values are determined by the value of t This type of relationship is called a function A functionis a relationship between two variables for which a unique value of thedependent variable can be determined from a value of the independent variable

  4. Using Graphing Calculator Pg 258 Enter Y1= 5 – x3 Press 2nd and table Enter graph

  5. Ex5.1, pg 264-265No. 40 g(t) = 5t – 3 • g(1) = 5(1) – 3 = 2 • g(-4) = 5(-4) – 3 = -20 – 3= -23 • g(14.1) = 5( 14.1) – 3= 70.5 – 3= 67.5 • g = 5 – 3 = - 3 = No. 51. The velocity of a car that brakes suddenly can be determined from the length of its skid marks, d, by v(d) = , where d is in feet and v is in miles per hour. Complete the table of values. Solution. V(20) Similarly put all values of d and find v

  6. Ch 5.2 Graphs of Functions (Pg 266)Reading Function Values from a Graph 2500 2400 2300 2200 2100 2000 1900 1800 P (15, 2412) f(15) = 2412 f(20) = 1726 Dow Jones Industrial Average Dependent Variable Q (20, 1726) Time Independent Variable 12 13 14 15 16 19 20 21 22 23 October 1987

  7. Vertical Line Test ( pg 269) A graph represents a function if and only if every vertical line intersects the graph in at most one point Function Not a function Go through all example 4 ( pg 270)

  8. Some basic Graphs • b = 3 a if b 3 = a Absolute Value Six Units Six Units -10 - 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 So absolute value of a number x as follows x = x if x > 0 - x if x< 0

  9. Graphs of Eight Basic Functions y = x2 g(x) = x3 f(x) = f(x) = f(x) = 1/ xg(x) = 1/x3 f(x) = x g(x) = -x g(x) = x

  10. No 15( pg 285) f(x) = x3 Guide point Guide point - 1 0 1

  11. 5.4 Domain and Range Enter y Enter window Press graph Range Domain

  12. STEP FUNCTION 5 4 3 2 1 Range 1 2 3 4 5 6 7 Domain

  13. 5.5 Variation Direct Variation Two variables are directly proportional if the ratios of their corresponding values are always equal The ratio = total price /number of gallons 20 10 5 10 15

  14. Other Type of Direct Variation • General equation, y = f(x) = kxn y = kx2 K> 0 = kx K> 0 y= kx3 K > 0 Inverse Variation y = n where k is positive constant and n> 0 y is inversely proportional to xn

  15. No 4, Ex 5.5 ( pg 309)The force of gravity( F ) on a 1-kg mass is inversely proportional to the square of the object’s distance (D) from the center of the earth F F= k/d2 ( k = constant of proportionality) a) Fd2 = k = 9.8(1)2 K = 9.8 b) F= 9.8/d2 substitute k 2 Distance Earth Radii Force (Newtons) Graph 8 6 4 2 Force 1 2 3 4 5 Distance d

  16. Pg 311, No 11The weight of an object on the moon varies directly with its weight on earth d) a)m w where m = weight of object, on moon and w= wt . Of object on earth m = kw m = 24.75 pounds, w = 150 pounds K = 24.75/150 = 0.165 m = 0.165w , substitute k b)m = 0.165( 120) = 19.8 pounds c) w= m/k = 30/0.165 = 303.03 pound Wt. on earth (W) Wt. on moon (m)

  17. Functions as Mathematical Models(Shape of the graph) Distance from Home Distance from Home Distance from Home Time Elapsed Time Elapsed bus wait walk Time Elapsed

  18. Example 5 , Pg 322 15 miles Gas Station Mall Highway 17 15 10 5 f(x)= - x + 15 x - 15 When 0 < x < 15 When x > 15 Miles from Mall 10 20 30 Miles in Highway

  19. Pg 323 y y 15 10 5 15 10 5 Miles from Mall x x 0 10 20 30 0 10 20 30 x – 15 < 5 10 < x < 20 x – 15 > 10 The solution is x < 5 or x > 25 Miles on highway

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