My English lesson ax + b Graph of the y = _____ function cx + d 1 If we shift the y = _ hyperbola by a translation vector v = (u ; w), the point P(u ; w) x corresponds to the origin and so it becomes the new centre of symmetry of the function. The straight-line y = w corresponds to the x-axis while the straight-line x = u corresponds to the y-axis. Its asymptotes are thus parallel to the Cartesian axes. y y=w P O x 1 In the figure, the graph s-coloured lines represent the function: 1 1 y w = _ that is y = _ + w x u x u x=u that can also be written as: xw uw + 1 y = ___________ x u BE CAREFUL! B It represents a function defined by a rational fraction that is the quotient of two polynomials (of first degree in x). Such a developed process can be reversed. It is always possible to find a translation ax + b that shifts a function y = _ where a and c do not simultaneously equal zero and cx + d 1 a d c b to a function such as y = _. Its graph is, therefore, a hyperbola with its x asymptotes parallel to the Cartesian axes. W are always referring to the We same Oxy system. Letters u and w specify the translation vector components: in each function they represent two numbers. Thus, the sum uw + 1 that is shown in the function expression, is a number. example e O Let us draw a graph of the function 3x + 4 y=_ x+1 Its domain is {x R | x 1} We can rewrite the numerator as 3x + 3 + 1 and, accordingly: 3(x + 1) 3x + 3 + 1 3x + 3 1 1 y = _ = _+ _ = _+ _ x+1 x+1 x+1 x+1 x+1 1 y = 3+_ x+1 1 The original function is thus related to the function y = _ by the translation: x x = x 1 {y = y + 3 The graph is symmetric to point P( 1 ; 3); the straight lines y = 3 and x = 1 are the asymptotes of the hyperbola (see below): y P 1 1 O 1 1 x Its graph is, therefore, a hyperbola with its asymptotes parallel to the Cartesian axes. 53

My English lesson