Bonsoir,
[tex]\textnormal{La forme canonique d'une fonction est sous la forme : $a(x-\alpha)^2 +\beta$ }[/tex]
[tex]\alpha = -\frac{b}{2a}[/tex] [tex]\beta = f(\alpha )[/tex]
[tex]a) \ f(x) = 3x^2 + 9x + 5[/tex]
[tex]a = 3 \ b= 9 \ c = 5[/tex]
[tex]\alpha = -\frac{b}{2a} \iff -\frac{9}{2\times3} \iff -\frac{9}{6} \iff -\frac{3}{2}[/tex]
[tex]\beta =f(\alpha ) \iff f(-\frac{3}{2}) =-\frac{7}{4}[/tex]
[tex]a(x-\alpha)^2 +\beta \iff 3(x -(-\frac{3}{2})^2 - \frac{7}{4} \iff \boxed{3(x+\frac{3}{2})^2 - \frac{7}{4}}[/tex]
[tex]b) \ f(x) = -2x^2 + 2x + 2[/tex]
[tex]a = -2 \ b = 2 \ c = 2[/tex]
[tex]\alpha = -\frac{b}{2a} \iff -\frac{2}{2\times(-2)} \iff \frac{1}{2}[/tex]
[tex]\beta =f(\alpha ) \iff f(\frac{1}{2}) =\frac{5}{2}[/tex]
[tex]a(x-\alpha)^2 +\beta \iff \boxed{-2(x-\frac{1}{2})^2 + \frac{5}{2}}[/tex]