Shifting a graph Use a shift to explain how the graph of

Question

ƒ (x) = \sqrt{- x^{2} + 8 x + 9}

is obtained from the graph of

g (x) = \sqrt{25 - x^{2}}

. Sketch a graph of ƒ.

Accepted Answer

Hello there today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Use a shift to explain how the graph H of X equals the square root of negative X squared minus six X plus 27 is obtained from the graph of P of X equals the square root of 36 minus X squared. Which of the following transformations applies, draw the graph of H of X Awesome. So it appears in this particular problem we're gonna be dealing with two separate graphs. We're gonna be dealing with the graph for H of X. And we're also going to be dealing with a graph of P of X. And we're trying to figure out how H of X is obtained from the graph of P of X. We're trying to figure out what exactly is happening during this transformation and we're trying to figure out what exactly trans what, what exact transformation is being applied to these graphs. And we're also trying to figure out a graph for H of X. So we're trying to solve for a few different things. So ultimately, we're trying to figure out what kind of transformation is gonna be taking place. So that's our end goal ultimately. So now that we know that we're trying to identify the transformation, which is our main goal. And then our second side goal is to create a graph which creating a graph might help us solve for our final answer too. Let's read off multiple choice answers to see what our final answer might be for the exact transformation that's taking place for this particular prompt. So A is shift the graph of P FX to the left side by three units. B is shift the graph of PF X to the right side by three units. C is stretch the graph of PF X vertically by a factor of three. And finally D is shift the graph of PF X to the left side by three units. OK. So first off in order to solve this problem, we have a bit of a predicament, we have a bit of a problem. So as we could see right now, in order for us to apply our transformations or even to figure out what kind of transformations are taking place, we're going to have to change the formatting of our graphs right now. So right now H FX and P FX are have a square root in the problem which we want to get rid of that square root. So we need to change our formatting. So in order to change the formatting, so we could better visualize and better see what kinds of transformations are taking place. We need to apply, completing the square method. So we need to recall and use and apply completing the square method in order to simplify our expression here. So let's take our H of X expression. So H of X is equal to the square root of negative X squared minus six X plus 27. And we need to be applying the completing the square method to this expression. So let me do that, we will get negative X squared minus six, X plus 27 is equal to negative parenthesis, X squared plus six X minus 27. And as you should recall, so I'm not gonna do every single step for you since you should be able to know, know from memory how to use the complete the square method. I'm gonna skip a few steps. So we need to keep it. We need to keep using the complete the square method down to its most simple form. And when we do that, we will find that it's equal to 36 minus X plus three to the power of two. So as we could see, this is, this is why it's very important to use completing the square method down to its most simple form. We could see what kind of transformation is taking place. We could see that we're taking X and we're adding three. So that means we're gonna be shifting our graph to the left. So we're gonna be shifting it to the left by three units because of this plus three, since we're adding three to X. So that means we're gonna be shifting our graph three units to the left. So to help us better visualize this problem, I've gone ahead and used some graphing software to graph our graphs for both P FX and H of X. And as we could see, we have H of X in green and we have P of X in blue. So our original graph, which we're gonna say is P FX because it's saying is obtained from the graph of P FX which is the square root of 36 minus X squared. And as you could see from our original graph, which is in blue, our shifted graph which is being shifted three units to the left in green, which is our H of X graph, which is our second graph that we were dealing with. And you could see that it's been shifted three units to the left, but it looks exactly the same as our original graph for P of X. The only thing that's different is that it's been shifted three units to the left. So let's make a note of this. So it's gonna be, and I'll do three and it'll have an arrow pointing to the left. So it's been shifted three units to the left. Awesome. So that will mean without a doubt, our final answer has to be multiple choice answer letter A which is once again shift the graph of P FX to the left side by three units. And that's it we've solved for this problem. So this problem is very easy to solve for as long as you have a strong understanding of what it means, like what all the little details mean. So you need to understand like when a graph is either gonna be stretched or compressed or shifted to the left or right. And it's always very helpful like if you're graph, like if the specific equation for the graph isn't written and it's the correct format which we had to use completing the square to get our graph in the right formatting in order to see what sorts of transformations are taking place. So that's very important you need to be able to identify like if you could look at the equation in the problem itself and see if you can figure out what transformations are taking place. And if you cannot figure it out, then you're probably gonna have to use completing the square or some sort of algebra to simplify the expression, to get in the proper formatting to do so. So with that said, this problem is very easy to solve. You just need to have a strong understanding of what happens to graphs, you know, they're either gonna be compressed or if they're gonna be stretched, et cetera. So you need to be able to identify transformation methods. And if anything I said was confusing, I would suggest to go back in your textbook and read about it a little bit more, watch more videos. But once you have a strong understanding of that, of those topics, this is a very easy problem to solve for and that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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Shifting a graph Use a shift to explain how the graph of $ƒ (x) = \sqrt{- x^{2} + 8 x + 9}$ is obtained from the graph of $g (x) = \sqrt{25 - x^{2}}$ . Sketch a graph of ƒ.