LINEAR MÔ HÌNH

CHAPTER CHAPTER 2 2 LINEAR FUNCTIONS AND LINEAR FUNCTIONS AND MODELS MODELS Temperature Reading Temperature Reading Temperature in Degree Celsius C Temperature in Degree Fahrenheit F 0 32 10 50 20 68 30 86 40 104 50 122 60 140 70 158 80 176 90 194 100 212 Scatter Plot Of The Data Scatter Plot Of The Data Average Rate Of Change Average Rate Of Change Temperature in Degree Celsius C Temperature in Degree Fahrenheit F Average Rate of Change ΔF/ΔC 0 32 10 50 20 68 30 86 40 104 50 122 60 140 70 158 50 32 18 1 8 10 0 10 . − = = − 68 50 18 1 8 20 10 10 . − = = − 86 68 18 1 8 30 40 10 . − = = − 104 86 18 1 8 40 30 10 . − = = − 122 104 18 1 8 50 40 10 . − = = − 140 122 18 1 8 60 50 10 . − = = − 158 140 18 1 8 70 60 10 . − = = − • The Average Rate of Change is Constant The Average Rate of Change is Constant Temperature in Degree Celsius C Temperature in Degree Fahrenheit F Average Rate of Change ΔF/ΔC 0 32 1.8 10 50 1.8 20 68 1.8 30 86 1.8 40 104 1.8 50 122 1.8 60 140 1.8 70 158 Scatter Plot Of The Data Scatter Plot Of The Data From the observations made in the previous slides we conclude that: ( ) ( ) 32 1 8 1 8 32F C . C OR F C . C = + = + For instance if the temperature reading is 37 ºC then the corresponding temperature reading in degree Fahrenheit will be: ( ) ( ) 37 1 8 37 32 66 6 32 98 6 F . . . F = + = + = o Population of Hypothetical City Population of Hypothetical City YEAR # OF YEARS SINCE 1990 POPULATION (IN THOUSANDS) 1990 0 110 1991 1 116 1992 2 122 1993 3 128 1994 4 134 1995 5 140 1996 6 146 1997 7 152 1998 8 158 1999 9 164 2000 10 170 Scatter Plot Of The Data Scatter Plot Of The Data Average Rate of Change of Population Average Rate of Change of Population YEAR #OF YEARS SINCE 1990 POPULATION (IN THOUSANDS) 1990 0 110 6 1991 1 116 6 1992 2 122 6 1993 3 128 6 1994 4 134 6 1995 5 140 6 1996 6 146 6 1997 7 152 6 1998 8 158 6 1999 9 164 6 2000 10 170 P t ∆ ∆ [...]... thousands Linear Functions A function of the form f ( x ) = ax + b Slope y-intercept where a and b are real numbers, is called a linear function of x  Sometimes we use y instead of f (x) y = ax + b Slope y-intercept Straight Lines & Their Equations A line in the rectangular coordinate plane can be represented algebraically by an equation of the form Ax + By = C An equation of this kind is called a linear. .. the form Ax + By = C An equation of this kind is called a linear equation in the variables x and y The graph of a linear equation is always a straight line If B ≠ 0 a linear equation can be written in slope-intercept form  A y = −  B Slope  C  x+ ÷ ÷  B  y-intercept Graphs Of Linear Equations 3x +5 y= x + 0y = 9 y 0 Vertical line x = Constant x -5 +7 x y 17 = 0x + y = -7 Horizontal Line y... intercepts, let y be zero and solve the resulting equation for x (0,4) x (-7,0) (0,-6) To find y intercepts, let x be zero and solve the resulting equation for y Graphing Linear Equations Using Intercepts Find the x- and y- intercepts of the linear equation 3 x + 5 y = 30 y-axis 3x (0,6) +5 y= 30 x-axis (10,0) Suppose that P = (x1, y1) and Q = (x2, y2) are two points on a non vertical line as shown below... − y1 Rise = x2 − x1 Run x Slope Of Straight Lines (9,10) y ,6) (5 Vertical line slope = Undefined (-5 ,3) x (5, -3) (-6,-7) (9,-8) ) -9 , 16 (- (15,-7) Horizontal Line Slope = 0 Point-Slope Form Of A Linear Function y ,6) (5 (-5 ,3) x (5, -3) (-6,-7) ) -9 , 16 (- (15,-7) Two non-vertical lines are parallel if and only if they have the same slope y 3x 3x 3x 3x 3x 3x +5 +5 y= 3x +5 y= -4 5 y= - 30 +5... -a a x Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other That is if one a b line has slope b then the other will have a slope of − a Piecewise -Linear Functions Schedule X of the 2005 Form 1040 instructions specifies the tax paid by a single taxpayer as shown below: Taxable income is Over… But not over… The tax is: Of the amount over… $0 $7,300 . since 1990. Linear Functions Linear Functions ( )f x ax b = + Slope y-intercept y ax b = + Slope y-intercept A function of the form where a and b are real numbers, is called a linear function. = An equation of this kind is called a linear equation in the variables x and y. The graph of a linear equation is always a straight line. If B ≠ 0 a linear equation can be written in slope-intercept. y + = Find the x- and y- intercepts of the linear equation (0,6) (10,0) 3 x + 5 y = 3 0 Graphing Linear Equations Using Intercepts Graphing Linear Equations Using Intercepts Suppose