In Exercises 45-52, use the graph of y = f(x) to graph each funct...

Question

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x) =(1/2) f(2x)

Accepted Answer

Welcome back. I am so glad you're here. We have a function F of X equals the square root of X. And we're asked to transform it to the function G of X where G of X is equal to three times the square root of two. X plus one. Alright, there are several things going on here for transformations. We've got this three in front of the radical. This too is being multiplied by the X. And then it has this plus one here after our X. So before we go into exactly what's happening for what those three transformations are. Let's zoom in here for a moment on what's going on inside the radical, What's happening directly to our function? And I want to factor out a two so that we've got two times X plus and then that one divided by two is one half and that's all still inside the radical and we still have the three being multiplied in front. So now let's look at those three transformations again. Now um when we have something being added to or subtracted from directly to the X. In this case we have this plus one half that indicates that we are going to shift horizontally and we are going to shift horizontally to the left Because it is positive and it's going to be to the left 1/2 of a unit. Then the next closest thing going on with this X is we have this two being multiplied still underneath the radical, so happening directly to our X. And because it's directly to our X. It indicates that there is a horizontal shift. And because the two is greater than one, it is going to indicate horizontal shrinking. The last thing happening is this three being multiplied on the outside of the radical. Not happening directly to our X. Function. So because it's not happening directly to it, that indicates that it's going to be a vertical transformation. And because that three is greater than one, it is going to be vertical stretching. So those are the three transformations taking place. Let's break down a little bit what this means. Looking at our graph here on the left, we know that this is the what's at the origin right now? The center of it at 00 is going to shift to the left one half of the unit. So it's not going very far, it's just going a little bit to the left and then we're going to have horizontal shrinking and vertical stretching. So that's going to look something like this. But not exactly. I'm just showing approximately what our G. Of X is going to look like. So we know that our new center because it didn't go up or down. It's going to be negative one half for our X. Value and still zero for our Y value because it didn't go up or down. We can also rewrite negative one half is negative 0.5 and zero. So that is the center we're going to be looking for. Let's look at our answer choices, -0.5 and zero. I'll rewrite that negative 0. zero. So let's look at a. That does have its new center negative 0.5 and zero. So A is a possibility. Let's look at B. That one has 01. That is not going to work. C. C. Has negative 0.5 and zero as well. So C. Is a possibility and D. Is positive 0.50. So D does not work. So what's the difference between A. And C. Well they have the Y intercept plotted A. S. Y intercept is at 1.732 and sees Y intercept is at 03. So let's go back up and be more specific with our horizontal shrinking in our vertical stretching. Let's find out exactly where our new white intercept is. So going back up to the top here we can solve for our Y intercept by plugging a zero in for X. So we'll have G. Of X equals three times the square root of two times zero plus one. And this will give us our y intercept so that we can determine which one is the correct graph. So two times zero, that's zero. So we've got the squared of zero plus one is 13 times a squared of one. Well the squared of one is one. So three times one. Well that's three. So our G. Of X. Or r Y value is three. So we also need an intercept at zero for our x. value and three for our y value now, which one of those two was it? While looking at our answer choices. The one that has a Y intercept at is answer choice C. So our correct answer is answer choice C. Well done, we'll catch you on the next one.

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In Exercises 45-52, use the graph of y = f(x) to graph each function g. g(x) =(1/2) f(2x)