Question

WILL GIVE BRAINLIEST AND 40 POINTS PLEASE HELP!!

Answer 1

Hmmm...

I think, f(x) is $f(x)=\left(\dfrac{1}{3}\right)^x$ not $f(x)=\dfrac{1}{3}x$.

The horizontal asymptote is y = -2.

Let $g(x)=\left(\dfrac{1}{3}\right)^{x+a}+b$.

We have two points on a graph ((-1, 1) and (0, -1). substitute:

$1=\left(\dfrac{1}{3}\right)^{-1+a}+b\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\ \text{and}\ a^{-n}=\dfrac{1}{a^n}\\\\1=\left(\dfrac{1}{3}\right)^{-1}\cdot\left(\dfrac{1}{3}\right)^a+b\\\\1=3\cdot\left(\dfrac{1}{3}\right)^a+b\qquad\text{subtract}\ 3\cdot\left(\dfrac{1}{3}\right)^a\ \text{from both sides}\\\\b=1-3\cdot\left(\dfrac{1}{3}\right)^a\qquad(*)$

$-1=\left(\dfrac{1}{3}\right)^{0+a}+b\\\\-1=\left(\dfrac{1}{3}\right)^a+b\qquad\text{subtract b from both sides}\\\\\left(\dfrac{1}{3}\right)^a=-1-b\\\\\text{substitute to}\ (*)\\\\b=1-3(-1-b)\qquad\text{use distributive property}\\\\b=1+3+3b\qquad\text{subtract 3b from both sides}\\\\-2b=4\qquad\text{divide both sides by (-2)}\\\\\boxed{b=-2}$

$\text{Susbtitute the value of b to}\ (*):\\\\-2=1-3\left(\dfrac{1}{3}\right)^a\qquad\text{subtract 1 from both sides}\\\\-3=-3\left(\dfrac{1}{3}\right)^a\qquad\text{divide both sides by (-3)}\\\\1=\left(\dfrac{1}{3}\right)^a\\\\\left(\dfrac{1}{3}\right)^0=\left(\dfrac{1}{3}\right)^a\iff a=0\\\\\text{Therefore we have}\ g(x)=\left(\dfrac{1}{3}\right)^x-2\\\\Translate\ the\ graph\ of\ f(x)\ 2\ units\ down.$