Precalc exponential function

Question

If f(x) = 3x – 2, find the exponential function which passes through f(6) and f(20). Round to the nearest

thousandth.

How do you do this question?? Do you use logs?

Answer 1

The problems can be generalized as follows: find the exponential curve which passes through the following points (x0,y0) and (x1,y1)

We assume that the exponential form is of the form F(x)=Ae^Bx which depends upon two parameters, A and B. We need to find the values of A and B such that the exponential curve passes through (x0,y0) and (x1,y1).

We impose that the curve passes through (x0,y0) by imposing F(x0)=y0=Ae^Bx0 we also impose that the curve passes through (x1,y1) by imposing F(x1)=y1=Ae^Bx1

We now take the ratio between these two quantities and we obtain

F(x1)/F(x0)=y1/y0=Ae^Bx1/Ae^Bx0 = e^B(x1-x0) . By taking the logarithms of both sides of the equality we obtain

log(y1/y0)=B(x1-x0) which implies B=(1/(x1-x0))*log(y1/y0)

If we multiply the above conditions we obtain y1*y0 = A²e^B(x1-x0) and hence A=sqrt(y0*y1/e^B(x1-x0))

In practice, in your case you have that the two points are given by (x0,y0)=(6,16) and (x1,y1)=(20,58)

and therefore you have that B=1/(20-6)*log(58/16)=(1/14)*log(3.625)=0.091989 and the value of A is given by

A=sqrt(58*16/e^B(20-6))=16

The exponential is therefore F(x)=Ae^Bx where A and B are the values evaluated here above.

Best,

Davide

Answer 2

Given function:

f(x) = 3x – 2

Calculating f(6):

f(6) = 3 • 6 - 2 = 16

Calculating f(20):

f(20) = 3 • 20 - 2 = 58

Exponential function in standard form:

y = ab^x + q

Note the exponential function goes through two points: (6, 16) and (20, 58).

Write the equations:

ab⁶ + q = 16

ab²⁰ + q = 58

We have 2 equations with 3 variables: "a", "b" and "q" - and we cannot solve this system: we must have 3 equations to solve this system of equations.

Let's assume that q = 0 and solve this system for this case.

ab⁶ = 16 → equation (1)

ab²⁰ = 58 → equation (2)

Divide the 2nd equation by the 1st equation:

(ab²⁰)/(ab⁶) = 58/16

b¹⁴ = 29/8

b = ¹⁴√ (29/8) ≅ 1.096

Calculating "a" from the 1-st equation:

a = 16/b⁶ ≅ 16/(1.096)⁶ ≅ 9.213

Resulting exponential function:

y = (9.213)(1.096)^x