Use shifts and scalings to transform the graph of

Question

ƒ (x) = x^{2}

into the graph of g. Use a graphing utility to check your work.

$g (x) = ƒ (2 x - 4)$

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. Apply the transformations on the graph of P of X equals X squared into the graph of H of X equals P of three X minus six. Check your work with the help of a graphing utility. Awesome. So it appears for this particular problem we're gonna be comparing our original graph, which our original graph is P FX equals X squared, which as you should start thinking about right now, this X squared is gonna give us a standard Parabola graph. So start thinking about Parabolas and we're comparing this original Parabola to this new Parabola because we're taking our original graph and we're gonna be using a new conditions. So we're gonna be transforming our original graph X squared into P of three X minus six. So what's gonna happen to our original graph when we have our new conditions of H of X equals P of three X minus six. So that's what we're trying to figure out is what's gonna happen to our original Parabola. So with that in mind, let's read off multiple choice sensors to see how our new graph is going to change. So A is horizontal stretch by a factor of three and the left shift by six units. B is horizontal stretch by a factor of six and right shift by three units. C is horizontal compression by a factor of one third and right shift by six units. And D is horizontal stretch by a factor of one third and right shift by six units. Awesome. So with that in mind, our first step to help us better understand what's going on in this problem, let's create a sketch of P FX equals X squared. So when we think of P FX is equal to X squared, like I mentioned earlier, this is our standard Parabola graph. So our standard Parabola drawing a very rough sketch here. So we have our Parabola which goes up and beyond forever and ever and ever. But its origin point is at 00. So that's our P of X equals X squared. So considering our original graph, we need to note that for H of X equals P of three X minus six, we need to note that since there's a three multiplied by X, let's underline it. Since there's a three multiplied by X, that will mean that we need to compress the graph of P of X equals X squared horizontally by a factor of one third. So we need to compress. So I'll draw an arrow so we need to compress by one third. So in order, so when we do that, when we compress horizontally by one third, and let's call our new graphs, it's we're gonna be applying some changes to P FX. So let me apply this change. So we're gonna be calling it Q of X now. So when we add our horizontal compression and we call this new graph Q of XQ of X is gonna be equal to P of three X or three multiplied by X because we're doing this horizontal compression. That is key, I can't say that enough because we're compressing the graph horizontally by three. So our next step now is we need to consider our second thing that makes this graph differently. So what makes her Parabola different in this case? So initially, we have our compression by one third, but we also have this minus six. So because we're subtracting six from X, that will mean we need to shift the graph which we're gonna be taking our Q of X equals P of three X. And we're gonna be shifting it to the right by six units. And when we do that, we will be able to get our final graph which our final graph that we're trying to graph to see how it changes is we're gonna be getting H of X is equal to P of and we're gonna take P of three X minus six because we're shifting it to the right by six units because we have this minus six or sub since we're subtracting six from our X. So with that in mind, to help us better visualize what exactly is happening in this problem. I've gone ahead and found a digitized graph. So I gone ahead and used some graphing software and I graphed our two little graphs here from our two different functions. And as we could see in red, we have our P of X, which is our X squared Parabola like we had in our sketch which starts at 00, the origin point. And we have our new graph which is H of X which as you could see, it's been compressed, meaning it's been squished is as you could see for our standard original Parabola, it's wide. So it's been stretched out and it's very open. Whereas our H of X graph is very narrow as you could see because it's been compressed, it's like someone squished both of the sides together to make it more narrow. And as you could see, it's been moved to the right by six units. And that's it. These kinds of graphs are very easy to do or this kind of problem is very easy to do as long as you understand what it, what it happens or what it means when it's like three multiplied by X or minus or plus some value after the X. So if it's either adding or subtracting from this X, what happens to your graph. And as long as you understand how these either like, you know, compress stretch or shift the graph. As long as you understand how that works, solving a problem like this is very easy to do. If any of this was confusing, I would suggest to watch more videos on this topic and to go back in your textbook to read more on this specific topic. But as long as you have a strong understanding of what, how to shift and how to either compress or stretch a graph, this problem is very easy to solve. So with that said, our final answer without a doubt has to be multiple choice answer letter C which once again is horizontal compression by a factor of one third and right shift by six units 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.

Use shifts and scalings to transform the graph of $ƒ (x) = x^{2}$ into the graph of g. Use a graphing utility to check your work.

$g (x) = ƒ (2 x - 4)$

Key Concepts

Function Transformation

Horizontal Shifts

Vertical Scaling

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