Consider parent function f(x)=e^x and the transformes function g(x)= -1/2e^3x+4, quantitatively explain the transformations to the parent function, f(x) that will produce g(x).

[tex]\text{Note:}\ g(x) = -Ae^{Bx-C}+D\\\bullet \quad \text{- represents a reflection across the x-axis}\\\\\bullet \quad \text{A represent a vertical stretch by a factor of A (shrink if}\ |A| < 1)\\\\\bullet \quad \text{B represents a horizontal stretch by a factor of}\ \dfrac{1}{B}\ (\text{shrink if}\ |B| >1) \\\\\bullet \quad \text{C represents a horizontal shift of C units}\ (\text{+ is right, - is left}) \\\\\bullet \quad \text{D represents a vertical shift of D units}\ (\text{+ is up, - is down})[/tex]

[tex]\text{Parent function:}\ f(x)=e^x\\\text{Transformed function:}\ g(x)=-\dfrac{1}{2}e^{3x}+4\\[/tex]

The transformed function has the following:

Negative: reflection across the x-axis

A = 1/2 vertical shrink by a factor of 1/2

B = 3 horizontal shrink by a factor of 1/3

C = 0 no horizontal shift

D = 4 vertical shift of 4 units up