In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transf...

Question

In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. h(x) = 2^(x+1) – 1

Accepted Answer

Hello, today we're gonna be graphing G of X as a transformation of F of X. So what we are given is F of X is equal to three to the power of X. Now what this is is a standard form of an exponential function. And the exponential function has a Y intercept at zero comma one and has a horizontal Asymptote and Y is equal to zero. Furthermore, this graph is gonna be approaching from negative infinity very close to the asy tote passing through zero comma one and then increasing exponentially. So this is going to be the rough shape of F of X. Now, what we want to go ahead and do is identify the transformations we need in order to go from F of X to G FX. So G FX is given to us as three to the power of X minus two plus four. At the end here, there's gonna be a total of two transformations we need. The first transformation revolves around the quantity X minus two as this is in the form of X minus H. And whenever you have an X minus H quantity that's going to indicate a horizontal shift of the graph X minus two is in the standard form of X minus H. And this is going to be where H is equal to two. So that means that we're going to have a horizontal shift of two units to the right of the graph. And the second transformation is going to revolve around the plus four. At the end of the function see plus four means that we're adding the constant plus K to the end of the graph. And whenever you're adding a constant to the end of the graph, that's going to indicate a vertical shift of the graph. What this means is that K is going to equal to four in this case. And we're going to have a vertical shift of four units up of the entire graph. So now that we've identified the transformations we need, let's go ahead and apply them to the F of X function and we're going to be using zero comma one as a reference point for our transformations. So let's go ahead and apply the first transformation which is going to be the horizontal shift of two units to the right. If we're, if we're applying this transformation, the 0.0 comma one is going to be shifted two units to the right. And that's going to give us the 0.2 comma one, the horizontal Asy tode doesn't change at all here. So we're still going to have a horizontal Asymptote of Y is equal to zero and the shape of the graph is not going to change. So we're still going to have this exponentially shaped graph. So that's gonna be the first transformation applied. Now, if we apply the second transformation, what we end up with is a graph where the 0.2 comma one is going to be shifted up for units. So the new point is going to be two comma five. And the thing about this graph is if we shift an exponential graph upward, a certain amount of units, we must also shift the horizontal ascent tode up that same amount of units. So since we're shifting the graph of four units, we're all, we're also going to have to shift the horizontal ascent tode up four units. So we're going to have a new horizontal ascent tode where Y is equal to four and the shape of the graph isn't going to change. So we're still going to have this exponential function that passes through the 0.2 comma five. And this is going to be the graph of G of X. Now that we have the graph of G FX, let's go ahead and identify the domain and range. The domain focuses on the least possible X value to the highest possible X value. So we're going to be reading this function going from left to right. But as you can see here, this graph is going to continue infinitely to the left. And as it increases, it's going to be moving to the right infinitely. So that means that the domain of the function is going to be all real numbers, which is represented as negative infinity to positive infinity. Now let's go ahead and identify the range. The range focuses on the least possible Y value to the highest possible Y value. See, keep in mind that we have horizontal ascent tote of Y is equal to four. That means that this graph is going to approach Y is equal to four infinitely but never actually equal four. So the range in this case is going to start at four, but it's not going to include the value of four. And this exponential function is going to increase in the Y direction infinitely. So the range is going to end with positive infinity. So the values of the range is going to be from negative four to positive infinity. And we're going to give both these values a parenthesis because we're not including the value of negative four and infinity doesn't have a finite value. So this is going to be the domain and range of the function G of X. Now that we've identified these three pieces of information, we just have to select the graph that accurately represents this and that graph is going to be option C. So the answer to this problem is going to be C. So I hope this video helps you in understanding how to graph this type of function and I'll go ahead and see you all in the next video.

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