How do you obtain the graph of

Question

How do you obtain the graph of $y=f\left(3x\right)$ from the graph of $y=f\left(x\right)$ ?

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. Given the function Y equals G of X. How would the graph of Y equals G of six X differ from the original graph? Awesome. So it appears we're comparing our original graph which our original graph is Y equals G of X and we're comparing it to Y equals G of six X. So now that we know that we're trying to figure out how G of X is different from G of six X, let us read, offer multiple choice answers to see how it's gonna be different exactly. So A is the graph will be compressed horizontally by a factor of 1/6 B is the graph will be stretched vertically by a factor of six C is the graph will be shifted six units upwards and D is the graph will be shifted six units to the right. Awesome. So in order to make solving this type of problem easier, let's break it up into five easy steps. So we'll have it, we could solve this problem in, in five easy steps. So our first step is we need to identify the transformation. So we'll just call, I'll just call it I for short. So we're trying to identify the transformation that is taking place for this particular problem. So the transformation from G of X, so we're dealing with G of X to G of six X involves the input variable X. So since X is being multiplied by six, that means that there's gonna be a horizontal scaling transformation. So let's do a capital H since we are dealing with a horizontal scaling transformation because we have our G of X. But then our next graph has a six multiplied by X. So that means because it's our variable X is being multiplied by six, we are dealing with a horizontal scaling transformation. Fantastic. So our next step for our second step is we need to, and I'm gonna call it D for short, which is the, determine the type of scaling. So now that we're determining the type of scaling, we need to note that sense. X once again is multiplied by a factor that is greater than one, which in this case, it is six, the graph of Y equals. And let's write this down. So we're taking our graph which is Y equals G of X. Wait oh It's so we're, so we're taking our original graph which is Y equals G of X and then we're gonna be changing it by Y equals G of six X. And this will mean that since the factor is greater than one, the graph which is G of six X will be horizontally compressed. So it's gonna be horizontally compressed. So let's do a capital C or compressed and we'll put it in quotation marks. So it's being compressed compared to our original graph of G FX by a factor of 1/6. So it's being compressed and I'll draw an arrow so it's being compressed by a factor of 1/6. So our next step which we're gonna call understand the compression. So we need to understand what's going on in this compression. We're gonna call it you for short. So as we should recall a note, a horizontal compression by a factor of 16 means that each point on the graph of Y equals G of X will move closer to the Y axis by a factor of six. And this is because as we should recall, the value of X that gives a particular Y value in G of X will now be reached six times sooner in G of six X. Awesome. So it's important to note and we'll write this down again. So Y equals G of X will be going to Y equals G of six X. So once again, it's important to note that sensors a horizontal compression by a factor of 16, each point on the graph for G of X will move closer to the Y axis by a factor of six. And this is because the value of X that gives a particular Y value in G of X will now reach six times sooner. So it'll now be reached six times sooner in our G of six times X. So we'll just put a six year in quotation mark. So we will remember that it's being reached by six times sooner. Nice. So our next step for four is we need to figure out the effect on the graph which I'll call this E for short. So how is this gonna affect our graph? So as we should recall and note the graph of Y equals G of six X will look like the graph of Y equals G of X, but it will be compressed towards the Y axis by a factor of 16, which will mean that if a point A comma B uh is on the graph of Y equals G of X where a comma B, you could consider that as like X comma Y. So XY coordinate will be on the graph of Y equals G of X. Then the point A divided by six comma B will be on the graph of Y equals G of six X. So let's make a note of this. So if we were to have our coordinate play, which is a comma B, so a comma B will be on the graph of Y equals G of X. So we're gonna say Y equals G of X, whereas comparing it on the other side of the spectrum here where it's a divided by six comma B will be associated with Y equals G of 66. Fantastic. So now our last step is we need to make our conclusion, which we'll call it C for short. So in conclusion, as we should recall a note based on all the information we've gathered together the graph of, as we should note, Y equals. So when you take Y equals G of six X will be horizontally compressed. So I'm gonna do HC or horizontally compressed version of the graph of. So horizontal compression of our original graph, which is Y equals G of X by a factor of so by 1/6. So by a factor of 1/6 so once again, quickly recap here. So our graph Y equals G of six X will be horizontally compressed version of Y equals G of X by a factor of 1/6 which will mean without a doubt, our final answer has to be multiple choice answer letter A which states that the graph will be compressed horizontally by a factor of 1/6. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

How do you obtain the graph of $y=f\left(3x\right)$ from the graph of $y=f\left(x\right)$ ?

Key Concepts

Function Transformation

Horizontal Compression

Graphing Techniques

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