The graph for the function f ( x ) = 3 x f\left(x\right)=3^{x} f ( x ) = 3 x is given below. Match the given function, g ( x ) g\left(x\right) g ( x ) , to its graph. g ( x ) = − 3 x + 2 + 1 g\left(x\right)=-3^{x+2}+1 g ( x ) = − 3 x + 2 + 1

In this question, we're asked to match the given function g of x to its graph. Here, my function g of x is negative 3 to the power of x plus 2 plus 1. Now we're also given the graph of the parent function of this, f of x is equal to 3 to the power of x, so we can do this based on transformations. So looking at my graph of g of x, the very first thing I notice about it is this negative sign. Now we know that having a negative on the outside of our function represents a tran or a reflection over the x axis, so that's the very first thing I'm going to look for in my new graph. Now my original function looks like this, so I'm looking for something that got reflected over the x axis and is in turn facing in a different direction. So looking at the graphs that I have down here, there's actually only one that's been reflected over the x axis and that is graph b. Graph a has not been reflected over the x axis. It's actually been reflected over the y axis, so we know that that is not correct. Now looking at graph c, no reflection happens, so I don't even wanna look at that graph either. So let's just verify that b has the other transformations that happen to our function as well. Now taking another look at our function, we have this x+2 in our exponent here. Now this represents our horizontal transformation. This is x minus negative 2. So this tells us that we would have a horizontal shift by 2 units to the left. Now, it looks like that did happen for graph b. And then we also have another shift, a vertical shift, up by 1 unit. Now looking at my original graph, f of x, it did get reflected over the x axis when I look at graph b down here. And then I also see that it did have those shifts occur, my horizontal shift and my vertical shift. So it looks like b is my correct answer here, and this is a graph of my function atg. Thanks for watching and let me know if you have any questions.