Use shifts and scalings to transform the graph of

Question

ƒ (x) = \sqrt{x}

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

$g (x) = 2 ƒ (2 x - 1)$

Accepted Answer

Below 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 the square root of X into the graph of H of X equals four multiplied by P of three X minus two. Check your work with the help of a graphing calculator. Awesome. So it appears for this particular pro we're gonna be taking our original function of P FX and we're gonna be applying transformations to it to obtain the graph of H of X equals four multiplied by P of three X minus two. Awesome. And we're also told that we can check our work with the help of a graphing calculator or some sort of graphing software. Awesome. So with that in in mind, let's read off our multiple choice answers to see what exactly is going to happen transformation wise. So A is horizontal stretch by a factor of three left shift by two units and vertical stretch by a factor of four B is horizontal stretch by a factor of three right shift by two units and vertical stretch by a factor of four C is horizontal compress by a factor of one third left shift by two units and vertical stretch by a factor of four. And finally, D is horizontal compress by a factor of one third right shift by two units and vertical stretch by a factor of four. Fantastic. So first off, let us create a sketch of our P FX graph. So I've gone ahead and used my graphing software to graph both of our functions. So looking at a, this is a screenshot of what I call using my graphing software. So as we could see, we have P FX our original function in blue and then we have our new function H of X in red. So as you could see using our original function as a starting point for what we're gonna use to transform our graph, we can see that three and let us underline it in red that we can see that three is multiplied to X and because three is multiplied to X, as we should recall, that will mean that we need to compress the graph of P FX equals square root of X horizontally by a factor of one third to obtain the following graph. And let's call this Q of X and Q of X is equal to P of three multiplied by X. And I'll do a little star here. So we're doing a horizontal compress. So I'm gonna say this HC so horizontal compress by a factor of one third. Awesome. So this is a horizontal compression HC of a factor of one third. And let's focus on our next variable here. So since two is subtracted from X so X minus two, so since two is subtracted from X, as we should recall, we need to shift the graph of Q of X is equal to P of three X by two units to the right now. Since it's minus two, that means we're gonna be shifting two units to the right. But if this was X plus two, then we'd be shifting our graph two units to the left, but it's minus two. So that means we're shifting two units to the right. And since we're shifting the graph two units to the right, we will obtain the following graph of H of X is equal to P of three X minus two, which is our final graph that we're ultimately, or almost our final graph that we're ultimately trying to solve for. But we have one more variable that we need to focus our attentions on which is this four multiplied by P. And since four is multiplied to this function, since it's multiplied to the function of P, that means we will have to stretch the graph of H of X equals P of three X minus two vertically by a factor of four to obtain our final graph. This is the graph that we're ultimately trying to graph for. So H of X is equal to four, multiplied by P of three X minus two. And once again, we need to do a vertical stretch. So we'll call this vs vertical stretch by a factor of four. So vs vertical stretch by a factor of four. Fantastic. So with that in mind, as we could see, so our original graph which is P FX starts at the origin point. But when we perform the transformations to our original graph, we will find that our new graph for H of X will be starting at some point like 0.6 or something means it's not quite at one. So that's our new position and it's the same exact graph as P FX, but it's been shifted around. So its location on the coordinate plane has changed. So essentially, once again to quickly recap what has happened to our original graph. So we have compressed our graph horizontally by a factor of one third, we've also shifted our graph two units to the right in order to obtain our new location. And finally, we've also had to perform a vertical stretch by a factor of four. So because we perform these cal these specific transformations, that will mean that our original graph has become the new location for H of X. And that's it we've officially solved for this problem. Hooray. So that means without a doubt our final answer has to be multiple choice answer letter D which states that horizontal compress by a factor of one third right shift by two units and vertical stretch by a factor of four. 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 $ƒ (x) = \sqrt{x}$ into the graph of g. Use a graphing utility to check your work.
$g (x) = 2 ƒ (2 x - 1)$

Key Concepts

Function Transformation

Horizontal and Vertical Shifts

Graphing Utilities

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