Application \( \begin{array}{lll}\text { Ecrire la forme canonique de } f(x) \text { dans chacun des cas suivants : } \\ \begin{array}{lll}\text { a) } f(x)=2 x+x^{2} & \text {;b) } f(x)=x^{2}-5 x+1 \text {;c) } f(x)=-x^{2}+x-2 \\ \text { d) } f(x)=2 x^{2}+\frac{3}{4} x+5 ; \text {;) } f(x)=(3 x-2)(x+3) ; & \text { f) } f(x)=-2 x^{2}-5 x+3 \\ \text { g) } f(x)=x^{2}+\sqrt{2} x+1 ; & \text { h) } f(x)=-\frac{1}{2} x^{2}+4 x ; & \text { i) } f(x)=x-2 x^{2} \\ \text { j) } f(x)=x(3 x-2)-2 ; & \text { k) } f(x)=\frac{1}{4} x^{2}-x-8 ; & \text { 1) } f(x)=-2 \sqrt{3} x^{2}-\sqrt{6} x\end{array}\end{array} \)

Pour la forme canonique, nous allons compléter le carré pour chaque fonction. a) \( f(x) = 2x + x^2 = (x + 1)^2 + 1 \) b) \( f(x) = x^2 - 5x + 1 = (x - \frac{5}{2})^2 - \frac{21}{4} \) c) \( f(x) = -x^2 + x - 2 = -\left(x - \frac{1}{2}\right)^2 - \frac{7}{4} \) d) \( f(x) = 2x^2 + \frac{3}{4}x + 5 = 2\left(x + \frac{3}{16}\right)^2 - \frac{71}{128} \) e) \( f(x) = (3x - 2)(x + 3) = 3(x - \frac{2}{3})(x + 3) - 6\) f) \( f(x) = -2x^2 - 5x + 3 = -2\left(x + \frac{5}{4}\right)^2 + \frac{41}{8} \) g) \( f(x) = x^2 + \sqrt{2}x + 1 = \left(x + \frac{\sqrt{2}}{2}\right)^2 + \left(1 - \frac{1}{2}\right) \) h) \( f(x) = -\frac{1}{2}x^2 + 4x = -\frac{1}{2}\left(x - 8\right)^2 + 32 \) i) \( f(x) = x - 2x^2 = -2\left(x - \frac{1}{4}\right)^2 + \frac{1}{8} \) j) \( f(x) = x(3x - 2) - 2 = 3\left(x - \frac{1}{3}\right)^2 - \frac{11}{3} \) k) \( f(x) = \frac{1}{4}x^2 - x - 8 = \frac{1}{4}\left(x + 4\right)^2 - 16 \) l) \( f(x) = -2\sqrt{3}x^2 - \sqrt{6}x = -2\sqrt{3}\left(x + \frac{\sqrt{6}}{4\sqrt{3}}\right)^2 + \frac{3}{8} \) Amusez-vous à retrouver les transformations !