What Transformations Convert e^t to a Damped Exponential Function?

[|<ings]

This is just to see if I remember? Please confirm, correct any errors, and answer the questions (q's in bold)

Homework Statement

What geometric transformations will "transform"

f(t) = e^t $$\stackrel{transformations}{\rightarrow}$$ $$\frac{mg}{b}$$ * (1 - e^-bt/m)?​

Homework Equations

f(t) = $$\frac{mg}{b}$$ * (1 - e^-bt/m)

= $$\frac{mg}{b}$$ - $$\frac{mg}{b}$$)e^-bt/m

The Attempt at a Solution

(1) vertical scaling? or horizontal?
f((-b/m)t) -> e^-bt/m
let the new f function be equal to g, another function
i.e. g(t) = e^-bt/m

(2) reflection across the x-axis, right?
-1 * g(t) = -e^-bt/m
let the new g function be equal to h, another function
i.e. h(t) = -e^-bt/m

(3) i think this one is the vertical scaling.
$$\frac{mg}{b}$$ * h(t) = -$$\frac{mg}{b}$$e^-bt/m
let the new h function be equal to i, another function
i.e. i(t) = -$$\frac{mg}{b}$$e^-bt/m

(4) vertical translation
i(t) + $$\frac{mg}{b}$$ = $$\frac{mg}{b}$$ - $$\frac{mg}{b}$$e^-bt/m

= $$\frac{mg}{b}$$ * (1 - e^-bt/m)

= f(t)

also, is the reflection across the y-axis [STRIKE]done by multiplying by -1 inside of the[/STRIKE] done like this?: f(x) $$\stackrel{reflection across y-axis}{\rightarrow}$$ f(-x). TNX

 

[Nino MMP]

Yes, the reflection across the y-axis is done by multiplying by -1 inside the function, f(x) → f(-x).