Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the graph when $A = ( 0,3 )$ and $B = ( - 3,24 )$. Round parameters to the nearest thousandth.

A) $y = - 3 e ^ { 0.693 x }$
B) $y = - 0.693 e ^ { - 3 x }$
C) $y = 3 e ^ { - 0.693 x }$
D) $y = 3 e ^ { 0.693 x }$
E) $y = - 0.693 e ^ { 3 x }$

Question

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the graph when $A = ( 0,3 )$ and $B = ( - 3,24 )$. Round parameters to the nearest thousandth.

A) $y = - 3 e ^ { 0.693 x }$ 
B) $y = - 0.693 e ^ { - 3 x }$ 
C) $y = 3 e ^ { - 0.693 x }$ 
D) $y = 3 e ^ { 0.693 x }$ 
E) $y = - 0.693 e ^ { 3 x }$

Quizplus · Accepted Answer

Using the two given points, we can form a system of equations: 
$$3 = a e^{b(0)}$$
$$24 = a e^{b(-3)}$$
Simplifying the equations, we get:
$$3 = a$$
$$24 = 3e^{-3b} \Rightarrow 8 = e^{-3b} \Rightarrow b = - \frac{\ln(8)}{3} \approx -0.693$$
Therefore, the exponential model that fits the points is:
$$y = 3 e^{-0.693x}$$
which is choice C.

Find the Exponential Model y=aebxy = a e ^ { b x }y=aebx

Find the Exponential Model $y = a e ^ { b x }$