Give a formula for the reaction illustrated using a vertical

Give a formula for the reaction illustrated using a vertical shift of an exponential function. The two points marked on the graph are A = (-1, -78) and B = (1, -6). The red horizontal line is given by y = -3, and is a horizontal asymptote of the function. y =

Solution

Solution:

Given Points are

A = (-1,-78) and B = (1,-6)

and y = - 3

If it is an exponential function, then it is in the form y = ae^bx + c

Since the horizontal asymptote is y= -3, c = -3

so plug the two points into y = ae^bx - 3

-78 = ae^(-b) - 3

-6 = ae^b - 3

-75 = ae^(-b)

- 3 = ae^b

multiplying those two equations, we get

225 = a², so a = 15 or -15.

Clearly it must be -15, since both values are negative and e^b is positive.

so -3 = -15e^b

1/5 = e^b

b = -ln 5

so y = - 15e^{(-x ln 5)} - 3

= - 15*5^(-x) - 3