In Exercises 67-80, begin by graphing the square root function, f...

Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x) = √(x+1)-1

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph H. Of X which is equal to the square root of X plus six. And then outside of the radical minus six. And we are to graph this as a transformation of the square root function which is F. Of X equals the square root of X. So the first step I'm going to take is to give us a little more room. So scrolling down and now I'm going to draw a rough sketch of a coordinate plane. So we'll have a vertical Y axis and a horizontal X axis. They meet together at the origin in the middle. Now for graphing F of X, if you recall exactly what the square root of X looks like on the graph, wonderful. If you don't you can always choose a few X values and you can plug those values in for X. In the function, find out the corresponding Y values and then plot those. So we're going to start there for our X values. I'm going to choose positive values because we have a domain restriction. Anything inside the radical here has to equal or has to be greater than or equal to zero. So I'm going to choose 41 and zero for my X values. Now, plugging those in to our function we have the square root of not X but four. The square to four is two. So we've got a point at four to looking at our coordinate plane, starting at the origin and we're going to move to the right four values for X. And then up to values for Y. So approximately here. Now plugging in one for X. That will be the square root of one. The squared of one is one. So now looking back at the origin, we're going to move to the right one unit and up one unit. So approximately here, now plugging in zero for X. That will be the square root of zero. The squared of zero is zero. So we've got one point right at the origin. Now, if we were to continue that, it starts at 00 and heads up toward positive infinity for both the X and Y axes, here's our F of X. All right now we need to plot our H of X. And we need to take a look at it and see what transformations are occurring. First of all, Where is our F. Of X. Inside of our H of X. And that's right here. The square root of X. Now, what are the transformations there are two of them were adding six here and subtracting six here. So what do each of those mean? Well, going with the order of operations for transformations of functions that we recall from previous lessons, we're going to start with what's closest to the X. And that's this plus six. That's still under the radical. That's happening directly to the X. Because it's under the radical with the X. So that indicates a horizontal shift. And in particular because we are adding six, It is a horizontal shift to the left six units. So I am going to start here. Let me label this 00 where R. F. Of X. Starts now we're going to take that starting point at 00 and shifted to the left six units. So this will be at -60. And that one's just temporary because we have one more shift to take care of. All right. So we've got this -6. But that is outside of the radical. It's not happening directly to our X. So that indicates a vertical shift. And in particular because we are subtracting six, it is going to shift our function down six units. So from that negative 60 point we're going down six units. So that will be approximately here at negative six. Negative six. So now it's going to look otherwise exactly like our F. Of X. So starting at negative six, negative six. It's going to go up in the same direction and there is our H. Of X. Right now that we've got that roughly plotted. We can pull that back up to the top and see if we have an answer choice that matches bringing this back up. And so again, we're looking for an H. Of X. That begins at negative six negative six. We look at our answer choices and this matches with answer choice D. Well done. We'll catch you on the next one

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x) = √(x+1)-1

Key Concepts

Square Root Function

Graph Transformations

Function Composition

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