Calculer le taux de variation d'une fonction $f$ entre deux points $a$ et $b$ revient à calculer le quotient $\frac{f(b)-f(a)}{b-a}$.\n\n1. Pour $f(x) = |x| + 2$, $a=0$, $b=6$ :\n\n$f(a) = |0| + 2 = 2$\n$f(b) = |6| + 2 = 8$\n\nTaux de variation = $\frac{8 - 2}{6 - 0} = \frac{6}{6} = 1$\n\n2. Pour $f(x) = |x - 3|$, $a=-1$, $b=3$ :\n\n$f(a) = |-1 - 3| = | -4| = 4$\n$f(b) = |3 - 3| = 0$\n\nTaux de variation = $\frac{0 - 4}{3 - (-1)} = \frac{-4}{4} = -1$\n\n3. Pour $f(x) = 2|x| + x$, $a=-5$, $b=-1$ :\n\n$f(a) = 2| -5| + (-5) = 2 \times 5 - 5 = 10 - 5 = 5$\n$f(b) = 2| -1| + (-1) = 2 \times 1 - 1 = 2 - 1 = 1$\n\nTaux de variation = $\frac{1 - 5}{-1 - (-5)} = \frac{-4}{4} = -1$\n\n4. Pour $f(x) = |x^2 - 3|x| - 2|$, $a=-2$, $b=2$ :\n\nCalculons $f(a)$ :\n$x^2 = (-2)^2 = 4$\n$|x| = |-2| = 2$\n$3|x| = 3 \times 2 = 6$\n$x^2 - 3|x| - 2 = 4 - 6 - 2 = -4$\n$f(a) = |-4| = 4$\n\nCalculons $f(b)$ :\n$x^2 = 2^2 = 4$\n$|x| = |2| = 2$\n$3|x| = 3 \times 2 = 6$\n$x^2 - 3|x| - 2 = 4 - 6 - 2 = -4$\n$f(b) = |-4| = 4$\n\nTaux de variation = $\frac{4 - 4}{2 - (-2)} = \frac{0}{4} = 0$