4 ESERCIZI La retta nel piano cartesiano 368 INVALSI 2014 Su un piano cartesiano sono rappresentati i grafici delle funzioni f e g definite nell insieme dei numeri reali e rappresentate dalle formule f (x) = 2x 5 e g(x) = 3x + 1. Aiutandoti anche con i grafici di f e g, indica se ciascuna delle seguenti affermazioni è vera (V) o falsa (F). V F a. f(x) = g(x) se e solo se x = 1,2 V F b. f(x) > 0 se e solo se x > 0 V F c. f(x) = 0 se e solo se x = 2,5 V F d. g(x) > f(x) se e solo se x < 1,2 y 7 6 5 4 3 2 1 4 3 2 1 O 1 1 2 3 4 5 6 7 x 2 3 4 PRACTICE WITH CLIL Bundle of straight line Given the equation: (*) 2x 3y + 1 + k(3x y 2) = 0 let us find out what is the set of geometric objects defined after the variation of k. 2x 3y + 1 + k(3x y 2) = 0 2x 3y + 1 + 3kx ky 2k = 0 (2 + 3k)x (3 + k)y + 1 2k = 0 (3 + k)y = (2 + 3k)x + 1 2k If (3 + k) 0 that is k 3 then: (**) y 2 + 3k 1 2k = (______)x + ______ 3+k 3+k What is obtained is the equation of a sheaf of lines, having as centre P, which angular coefficient has direction: 1 2k 2 + 3k and intersection with the axis of ordinates in (0 ; ______). m = ______ 3+k 3+k To know the coordinates of point P (centre of the sheaf) it is sufficient to choose two values for k and put in a system the two straight lines obtained. For example: if k = 2 x=1 y = 4x + 5 x = 4x + 5 5x = 5 1_ {y = x {y = x {y = x {y = 1 _ if k = 2 The difference between the set of equations (*) and (**) is given by the fact that (*) also considers the straight line x = 1 (when k = 3) which is not included in (**) because this last sheaf is defined only if k 3. Exercise 1. Write, in the form (*) all the straight lines of the sheaf having its centre in: P(1 ; 2); P( 2 ; 3); P(0 ; 3) 179