RELAZIONI E FUNZIONI PRACTICE WITH CLIL Quadratic polynomials In order to draw a graph of a fourth degree polynomial, we should know how to transform graphs starting from the curves in their elementary forms, in addition to factoring, and the first and second degree equations. Quadratic equations with a > 0 can be obtained by translation (or expansion) of one of the following three (to make it simpler, we have chosen them symmetric to the y-axis; we can prove, in fact, that each quadratic equation is symmetric to one line parallel to the y-axis): x4 x 2 x4 x4 + x 2 y 2 O 2 x By observing these graphs (a = 1), we can note that a quadratic function can have: no intersection with the x-axis (it is sufficient to translate upwards each of the three types); two congruent intersections with the x-axis (translated along the x-axis of the blue type); two distinct intersections with the x-axis (a function translated downwards of each three types; four distinct intersections with the x-axis, or two distinct and two congruent, or two distinct couples of congruent (a function translated upwards of the green type); four congruent intersections (a function translated along the x-axis of the red type). Quadratic equations with a < 0 can be obtained by translation (or expansion) of one of the following three (a = 1): y O 2 2 x x4 x4 + x 2 x4 x 2 Exercises 1. Which information can you deduce from these graphs? 2. Functions that are equalling to 0 are binomial equations of the fourth degree. By which graphs are they represented? 3. Functions that are equalling to 0 are biquadratic equations of the fourth degree. By which graphs are they represented? 296