How do you graph $f(x)=(1/3)^x-3$ by plotting points?

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Answer 1 · 2017-07-04T02:36:56

See below

$f(x) = (1/3)^x - 3$

Before we start plotting points, let's first get an idea of some of the characterics of $f(x)$

$lim_(x->+oo) f(x) = 0 -3 = -3$
We should note that $f(x) -> -3$ very rapidly.
I.e. We wont need very many points $x>0$

$lim_(x->-oo) f(x) = lim_(x->+oo) 3^x-3 = +oo$
Again $f(-x) -> oo$ quite rapidly.

$f(0) = (1/3)^0 -3 = 1-3 =-2$

So, $(0, -2)$ is a point on our graph

$f(x) =0 -> (1/3)^x = 3$

$3^(-x) = 3^1 -> x=-1$

So, $(-1, 0)$ is another point on our graph.

Calculating other points:

$f(1) = 1/3-3 approx -2.667$

$f(2) = 1/9-3 approx -2.889$

$f(3) = 1/27-3 approx -2.963$

$f(-2) = 9-3 =6$

$f(-3) = 27-3 =24$

We can use the chracteristics and calculated points of $f(x)$ to plot a graph as produced below.

graph{(1/3)^x-3 [-7.9, 7.9, -3.935, 3.97]}

How do you graph f(x)=(1/3)^x-3f(x)=(1/3)^x-3 by plotting points?