The last image is the graph of [tex] y = 3^x [/tex]
In fact, it is an increasing exponential function, and it passes through the points [tex] (0,1) [/tex], which reflects the fact that [tex] 3^0=1 [/tex] and [tex] (1,3)[/tex], which reflects the fact that [tex] 3^1=3 [/tex].
Now, [tex] y = 3^{x-4} [/tex] is a child of the parent function we just described. Precisely, it is the result of the transformation [tex] f(x) \to f(x-4) [/tex]
In general, every time you perform a transformation like [tex] f(x) \to f(x+k) [/tex], you translate the graph horizontall, k units to the left if k is positive, and k units to the right if k in negative.
Since in this case [tex] k = - 4 [/tex], we have a horizontal translation of 4 units to the right.
So, the correct option is the third one, because:
- The first graph is the parent function translated 4 units to the left
- The second graph is the parent function translated 4 units down
- The third graph is the parent function translated 4 units to the right
- The fourth graph is the parent function
