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 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.

6. Exponential and Logarithmic Functions

Graphing Exponential Functions

Precalculus

6. Exponential and Logarithmic Functions

Graphing Exponential Functions

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Multiple Choice

The graph for the function $f\left(x\right)=3^{x}$ is given below.

Match the given function, $g\left(x\right)$ , to its graph. $g\left(x\right)=-3^{x+2}+1$

Verified step by step guidance

Identify the function type: The function g(x) = -3^(x+2) + 1 is an exponential function with a negative base, which means it will have a reflection across the x-axis compared to a positive base exponential function.

Consider the transformation: The term (x+2) indicates a horizontal shift to the left by 2 units. The +1 outside the exponential function indicates a vertical shift upwards by 1 unit.

Analyze the graph behavior: Since the base is negative, the graph will decrease as x increases, and the horizontal and vertical shifts will adjust the position of the graph accordingly.

Compare with the given graphs: Look for a graph that shows a decreasing trend, has been shifted left by 2 units, and shifted up by 1 unit.

Match the graph: The correct graph will show the characteristics of a reflected exponential decay, shifted left and up, which should match one of the provided graphs.