Making a prediction with a line.This is done by taking two points and drawing a linethrough them. We take the corresponding years and population and put them into the linear equation.
It will look somewhat like this.1850 is referred to as year 0 because it is the year we started with. The points i used for this are year 100(1950) and year 150(2000). These are used as the X values with the populations as the Y values.
y1-y2 = Slope y-y1=m(x-x1) x1-x2 68377000-82797400 =288408=Slope 100-150 y-68377000=288408(x-100) y-68377000=288408x-28840800 +68377000 +68377000 y=288408x+39536200We then plug in the year that we want to predict the population. 2061=211 y=288408(211)+39536200 y=100390288Predicted population of Germany in 2061~100,390,288
Making a prediction with a QuadraticWe need to chose 3 points for this prediction. we will plugthese points into the equation ax^2+bx+c and use them to ﬁnd the values of a b and c. We also need to write the populations in a much more manageable way. We write them in scientiﬁc notation, moving the decimal over 7 places.
x1,y1 0,3.5 3.5=a(0)^2+b(0)+c 3.5=a(0)^2+b(0)+c x2,y2 100,6.8 6.8=a(100)^2+b(100)+c 3.5=0+0+c x3,y3 110,7.2 7.2=a(110)^2+b(110)+c 3.5=c 6.8=a(100)^2+b(100)+3.5 7.2=a(110)^2+b(110)+3.5 3.7=12100a+110(.033-100a) 3.3=10000a+100b 3.7=12100a+3.63-11000a 100 3.3=10000a+100b .07=1100a 3.7=12100a+110b .033-100a=b 1100 3.3=10000(.00006)+100b a=.00006 b=.027 3.3=.6+100b 2.7=100b Our Equation. y=.00006x^2+.027x+3.5 100Now we plug in the year we want to predict (211) once again as the x value. y=.00006(211)^2+.027(211)+3.5 Predicted population: y=.2.67126+5.697+3.5 118,682,600