Answer:
Graph has been shown in the attachments.
Step-by-step explanation:
We have to graph the function [tex]y=\left(\frac{2}{3}\right)^x-2[/tex]
It represents an exponential function. Let us find the x and y intercepts,
For x-intercept, we plug y = 0
[tex]0=\left(\frac{2}{3}\right)^x-2[/tex]
[tex]2=\left(\frac{2}{3}\right)^x[/tex]
Take log both sides, we get
[tex]\log 2=\log(\left(\frac{2}{3}\right)^x)[/tex]
[tex]x=-1.71[/tex]
For y-intercept, we plug x=0
[tex]y=\left(\frac{2}{3}\right)^0-2[/tex]
[tex]y=-2[/tex]
Therefore, the graph must passes through the points (0,-2) and (-1.71,0)
The horizontal asymptote of the graph is given by
[tex]y=\lim_{x\rightarrow \infty }g(x)[/tex]
[tex]y=\lim_{x\rightarrow \infty }\left(\frac{2}{3}\right)^x-2[/tex]
[tex]y=-2[/tex]
Hence, using these information, we can easily graph the function.
