In Exercises 51–54, graph the given square root functions, f a...

Question

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g (x) = √(x + 2) (x = = −2, −1, 2, 7)

Accepted Answer

Welcome back. I am so glad you're here. We are given this function F of X equals the square root of five X. And its friend and sort of relative G of X equals the square root of five X plus eight. And were given some domain restrictions for each of them. They have to be where X is greater than or equal to zero but less than or equal to 16 for both of them. So what we are to do with our two functions is to graph them. And then we're supposed to compare the graph of G of X with the graph of F of X. All right, So let's get started on graphing them for graphing F of X. We can choose to plot a few points. And remember for our X values they need to be between zero and 16. So, I'm going to start with zero. And then I'm also going to choose five and just to have a few different points. And I am also going to draw a Cartesian plane. I am going to draw a vertical Y axis and a horizontal X axis. And yes, I'm kind of skewing it a little bit just because I've done this ahead of time and I want it to be a little bit easier to see what we're doing. So now let's graph our points for a five X. We're going to plug in zero for X. So we have the square root of five times not X now, but zero and well five times zero is zero. So we have the squared of zero in the squared of zero is zero. So our first point is right at the origin 00. So we look at our graph and we're going right to the origin at 00. There's one point. Alright now let's graph where X equals five. We have the screwed of five times no longer acts but 55 times five is 25. So we have the square root of 25 while the square root of 25 is five. So we look at our graph and we're going to move to the right five for our X value and we're going to go up five for our Y. Value. And that's approximately here. Alright now let's work on 10 for our X. Value. We'll take the square root of five times not X. But 10. And we could re write this, this could be the square root of five times 10 is five times two. So five times five times to. Well that is the same as well. The square root of five times five is five. So five out in front and then leave that square root of two inside the radical. So the corresponding y value here would be five square root two. So looking at our graph will go to the right 10 from the origin for our X value and we'll go up five square to for our Y value which is just a little bit higher than our five. So we can draw this and it looks something like that. I'm just going to move my dot there. Okay. There we go. We've got our F. Of X. I'll label that and now we get to label our G of X. Well our G. Of X is very similar to our F. Of X. Really. The only difference is this plus eight afterward. So this is a shift that we can identify. We recall from previous lessons that we know this plus eight isn't happening directly to the X. It's happening outside of the radical. So this is going to be a vertical shift and because it's adding eight, it's going to be a vertical shift of going up eight units. So whereas ffx started at 00 at the origin R. G of X is going to start at 08. It's going to go up eight. So because that was the only shift taking place, it will otherwise look just like our F. Of X. So here's our that's not how you right G. Of X. Here's our G. Of X in blue. Okay. We have graphed this and now we can look at our different answer choices to see which ones make sense. So I'm going to move this down a little bit and we can look at A. And B first. Okay. A. Well F of X does start at 00 but that one is going down to the right. So A does not work. Let's look at B. F of X does start at 00, but it's going down to the right, so B does not work. All right. Let's check out C and D bring this down. We can see them both. All right. Okay. Looking at answer choice C F of X does start at 00 and goes up to the right answer choice D. F of X does start at 00 and goes up into the right. But the G of X is different for answer choice. C. R G of X is eight units above our F of X and for answer choice D. It's eight units below. So, we know answer choice D. Doesn't work. Let's see what the description is comparing them for answer Tracy. It says the graph of G is the same as the graph of F shifted eight units upwards. Well, that is exactly what we're looking for. So, it's answer choice C. Well done. We'll catch you on the next one.

Key Concepts

Square Root Function

Domain and Range

Transformations of Functions

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