Question

12
soit une fonction exponentielle dont la règle est de la forme
$f(x) = ac^{b(x - h)} + k$, où $0 < c < 1$.
quelles sont les valeurs possibles des paramètres $a$ et $b$ ?
a) $a > 0$ et $b < 0$
b) $a > 0$ et $b > 0$
c) $a < 0$ et $b > 0$
d) $a < 0$ et $b < 0$
graph of the exponential function is shown with x and f(x) axes, the curve is in the fourth quadrant-like area (x positive, f(x) negative? or as per the images curve shape)

Explanation:

Step1: Analyze base behavior

Given $0 < c < 1$, $c^t$ decreases as $t$ increases.

Step2: Analyze function trend

The graph shows $f(x)$ increases as $x$ increases. For $f(x)=ac^{b(x-h)}+k$, this means $b(x-h)$ must decrease as $x$ increases, so $b < 0$.

Step3: Analyze vertical direction

The graph is below the horizontal asymptote (dashed line), so $a < 0$ (since $c^t > 0$ for all $t$, negative $a$ flips the curve downward).

Answer:

d) a < 0 et b < 0