: The graph of the function ????=????(????) represents a transformation of the graph of ????=????(????). Determine the equation of ????(????) in the form ????=????????(????(??????))+????.

Name: the graph of the function ygx represents a transformation of the graph of yfx determine the equation of gx in the form yafbxhk
Uploaded: 2021-10-15
Duration: 7 min 47 s
Channel: Numerade
Description: : The graph of the function ????=????(????) represents a transformation of the graph of ????=????(????). Determine the equation of ????(????) in the form ????=????????(????(??????))+????.

Question

David Mccaslin · Accepted Answer

So in this problem we have the function Y equals F of X. And it&#x27;s been translated and transformed to the function Why equals G F X. And were asked right, the function then G F X in the form a F of B times x minus H plus. Okay, Okay. So first of all, let&#x27;s do this. If we look at the function F of X up here, right? It&#x27;s it hits here at -2 goes up deposit for right now goes up positive for goes back down at Looking to be one Because these are in increments of two. It was meant one and back up to four. one where X is one. The other graph G of X goes down first starting from the left and then up and then back down. So it&#x27;s like the mirror image of each other, aren&#x27;t they? Okay. So that tells me that A is negative because I need to flip F of X over. All. Right. If I flip F of X over, then what happens instead of hitting? So it starts here at minus two and instead of going up to positive for it would go down to negative for right. So it would go this way and then that way and in that way. Okay, So that starts to get me there now. Okay, what else do I notice? Well, I notice that the left most point right Is still at zero for why? Right. And The lowest points are at -4 for why? Which is still happening? So there&#x27;s really not a shift up or down. So that tells me that K is zero, right? I don&#x27;t have a shift since K is adding to the whole function, I don&#x27;t have a shift up or down on this at all going on. Okay. Now that just means that we got to get this spread out, right? We got to get it to go, oh that way, don&#x27;t we back and forth? Okay, well, so when X is zero, I&#x27;m trying to get Instead of -1, I&#x27;m trying to get I&#x27;m sorry sir, my -2. No, When X is -2, I need that to be -7, don&#x27;t i? And still get zero for this thing. And so what did I do? I shifted this point on X. I shifted it left five. Right, mm, Okay, but then this point here is that three? I needed to become 13, so I need it to be shifted 10 that direction, Don&#x27;t I? Mm. Okay, well, okay, so we&#x27;re looking at these shifts and all and let&#x27;s begin by noticing at this point that uh huh we shift at this point right here over three. Right, So when X is three, I&#x27;m doing f of zero, aren&#x27;t i? Okay, so that right there tells me so far that G fx is minus F of X minus three. Right, okay. Is zero. Of course I know that much so far. Now we know that F of F Of Zero, right, it&#x27;s -4. So whatever we put in here for B will still make this true for this point. So let&#x27;s look at this shift right here. Okay, so we need to go from -2 Over 2 -7. In other words I put -7 in for X. Right? So I have G -7 his minus F of something times -7 -3. Right? Not minus 30 minus three. Okay well so that&#x27;s minus F of B Times -10 in it. And I want To get f of -2. So let&#x27;s put b Go to to over 10 in other words 1/5 and watch what happens. So now I have G fx is minus F of 1/5 times x -3. So now let&#x27;s look at what happens, I put -7 in there for X. I get -10 Divided by five is -2 and F of -2 is zero. So that ties now, guess when this point tied up We have when X is three I get zero, Next -3 would be zero. F 0 is 4 -4. So I just tied that point. So let&#x27;s try this one now. eight. Well, So that means 8 -3 is five times 1/5. That&#x27;s one. So it&#x27;s F of one. What F of one is zero? So that ties? Let&#x27;s check this one. Last point 13. So X is 13, 13 months, three is 10, 10 divided by five is too So it means f of two. Well f of two is is for Negative, gives me -4. And so there is our function with all the shifts on it, all the transformations.

Joseph Lentino · Answer

So Russian 42 we&#x27;re looking at right now is a transformation of a rabble. And in this particular case, we can see that we&#x27;re gonna be shifting it to the left to we&#x27;ll be shifting it up. One The easiest way to get a nice picture A nice, perfect transformation grab is to start with the parent function. And I&#x27;m gonna pick a few points off the parent forms. Parent function here would be y equals X squared. So negative to negative. 101 and two. So 41014 When you plug in my X values, the transformations that are happening, anything on the outside will affect the outcome. So in this case, even add ones all of my output values and I&#x27;m going to subtract chew because remember, everything inside of the function will be the opposite value. When I should track, I&#x27;m gonna get negative for negative three negative to negative one and zero. The outside columns that are left over are the new points that we will grab. So when I graft these points negative for five negative three to negative 211 chew and 05 You&#x27;ll see that we have That original parable has shifted to the left

Yifan Xu · Answer

All right. So we have Why equals half of X Plus three are So this means that we shift the original function effort, fax to the left by three units.

Joseph Lentino · Answer

So Question 57 is going to ask us to do a little bit more work because we&#x27;re working with the reciprocal function one over X plus three plus two. Now, one of the things that we have to remember is my parent function, which is why equals one over X. Okay, this will give us a graph that will have Assam toads on both axes. So I have to use a few more points in order to graft this a little bit nicer. So the parent function is why equals one over X. So the points I&#x27;m going to choose right here are going to be negative, too negative. One negative. 1/2 zero, 1/2 one into. So it&#x27;s a few more points than normal. And when I take the reciprocal of negative too, I get negative 1/2 The reciprocal of negative one is negative one. The reciprocal of negative 1/2 is negative. Two. The reciprocal of zero is going to be undefined. Which is going to tell us where our acid it will be a vertical aspecto The reciprocal of 1/2 is to the reciprocal of one has won. The reciprocal of two is 1/2. Now, anything that&#x27;s inside the function will affect the input. Notice that in this case, I&#x27;ve got X Plus three, which wasn&#x27;t going to shift the graph. Three units to the left, Which means when I affect the input, which is my ex values, I&#x27;m going to subtract three units. That&#x27;s why move it left three units, which will be negative. Five negative. Four negative. 3.5. Negative three. Okay, I&#x27;m going to have, um negative 2.5. Negative too. And negative one. And now my ex values have been changed. Now these the six points on the outside will be points. Sorry about that. Negative four negative. 3.5. Negative. 2.5. Negative two and negative one. But the the wares undefined X equals negative three will actually be my vertical ass in tow. Now, we need to manipulate our wise We&#x27;re gonna affect our y values by adding to, um, noticed that that will also mean that I&#x27;m going to shift up the horizontal assam toe to units. So in this case, when I add to to all of my y values, I&#x27;m gonna have 1.5 one zero still undefined. Can&#x27;t undefined post. You is still undefined for three and 2.5. And so when I go to graph this, I know that there&#x27;s going to be a vertical ascent haute and negative three. I also know because I&#x27;m shifting it up two units that there will be a horizontal Assam tote at two. And when I graphed the rest of these points 12345 1.5. Negative for one, 3.5 0 We can see our reciprocal function here and then I graphed Negative 2.5 3 Sorry, negative. 2.5 4 Apologize. Negative. 2.5 4 Negative 23 Negative 1 2.5 and we can see.

Joseph Lentino · Answer

so Question 39 using transformations We&#x27;re going to graph this proble now. I always like to identify using the transformations using the thick parent function. So in this case, the parent function is why equals X square. So if I choose some X values negative to negative 1012 I know that there are the points for 1014 on the graph, and the easiest way to do the transformation is to effect the inputs and the outputs of my original function. So in this particular case, because this is happening on the inside, this is going to be a shift left one unit. So I left and right is my input value. So in this particular case, I&#x27;m gonna take each of my input values and subtract one so negative three negative to negative 10 and want. By doing that, I mean end up with negative three fours, a new point negative to one as a new point. Negative 10 is a new point 01 and 14 and I noticed I didn&#x27;t affect the outputs because there was nothing happening outside of my function sketching this graph, you&#x27;ll see it looks almost identical to the original picture just shifted over to the left one unit

Discussion

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Continuity - Overview

Continuity - Example 1