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 “

Ch 5.1 (pg 251) Definition and Notation Example – To rent a plane flying lessons cost $ 800 plus $30 per hour Suppose C = 30 t (t > 0) When t = 0, C = 30(0) + 800= 800 When t = 4, C = 30(4) = 920 When t = 10, C = 30(10) = 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 function is a relationship between two variables for which a unique value of the dependent variable can be determined from a value of the independent variable tc (t, c) (0, 800) (4, 920) (10, 1100) Table Ordered Pair

Using Graphing Calculator Pg 258 Enter Y1= 5 – x 3 Press 2 nd and table Enter graph

Ex5.1, pg No. 40 g(t) = 5t – 3 a)g(1) = 5(1) – 3 = 2 b)g(-4) = 5(-4) – 3 = -20 – 3= -23 c)g(14.1) = 5( 14.1) – 3= 70.5 – 3= 67.5 d)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 d v

Ch 5.2 Graphs of Functions (Pg 266) Reading Function Values from a Graph October 1987 Dow Jones Industrial Average Dependent Variable P (15, 2412) Q (20, 1726) Time Independent Variable f(15) = 2412 f(20) = 1726

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)

Some basic Graphs b = 3 a if b 3 = a Absolute Value Six Units So absolute value of a number x as follows x = x if x > 0 - x if x< 0

Graphs of Eight Basic Functions y = x 2 g(x) = x 3 f(x) = f(x) = f(x) = 1/ x g(x) = 1/ x 3 f(x) = x g(x) = -x g(x) = x

No 15( pg 285) f(x) = x 3 Guide point

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

STEP FUNCTION Range Domain

5.5 Variation Direct Variation Two variables are directly proportional if the ratios of their corresponding values are always equal Gallons of gasoline Total Price Price /gallons 4$ /4 = $ /6= $ /8 = $ /12 = $ /15 = 1.15 The ratio = total price /number of gallons

Other Type of Direct Variation General equation, y = f(x) = kx n y= kx 3 K > 0 y = kx 2 K> 0 = kx K> 0 Inverse Variation y = n where k is positive constant and n> 0 y is inversely proportional to x n

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/d 2 ( k = constant of proportionality) a) Fd 2 = k = 9.8(1) 2 K = 9.8 b) F= 9.8/d 2 substitute k Distance Earth Radii Force (Newtons) Distance d Force Graph

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

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

Example 5, Pg 322 Gas Station Mall Highway miles Miles in highwa y Miles from Mall Miles in Highway Miles from Mall f(x)= - x + 15 x - 15 When 0 < x < 15 When x > 15

< x < 20 x – 15 < Miles on highway Miles from Mall x – 15 > 10 The solution is x 25 x y x y Pg 323