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. g(x) = √(x+1)

Accepted Answer

Hello, today we're going to be graphing H of X as a transformation of F of X. So what we are given is F of X is equal to the square root of two X. And we are told that H of X is equal to the square root of the quantity two X minus six. So before we graph H of X, we need to get an idea of what the shape of F of X is going to look like. Currently FX has a product inside of the square root as we have two times X. So we can separate this as a product of two square roots. This is going to give us the square root of two times the square root of X. And this is going to be F of X simplified. Now F of X uses the square root of X. So we're going to need to figure out or understand what the original shape of the square root of X function looks like. See for the parent function Y is equal to the square root of X, we're going to have the graph on the axis whose starting point is at the origin zero comma zero. And the shape of this graph is going to be a curve that stretches off to the right. So this is the traditional form of the square root of X. But here F of X has that graph but with the coefficient of the square root of two. So in order to go from the traditional square root of X graph to F of X, we're going to need one transformation which involves the coefficient of two. See the coefficient of two produces a value that is going to be greater than one. And that means that the shape of this graph is going to have a vertical stretch. So that means if we want to graph Y of X, we're going to take the traditional square root of X graph and we're going to stretch it vertically. So a vertical stretch means this graph is going to be taller now. So this is going to be the shape of F of X. Now that we know the shape of F of X, let's go ahead and figure out what transformations we need in order to go from F of X to H of X. The first thing we're going to do is to simplify H of X. And we can do that by first factoring out of two from the quantity two X minus six doing this is going to give us the square root of two times the quantity X minus three. And since we have a product inside of the square root. We can go ahead and separate this into a product of two square roots. This is going to give us the square root of two times the square root of X -3. Now, a similarity between H of X and F of X is that they both have the coefficient of the square root of two. So the shape of the graph is going to stay the same. But there's going to be one transformation. We need to go from F of X to H of X. And that's going to come from the quantity X minus three, C X minus three is of the form X minus H. And whenever you have a quantity that's in the shape of X minus H, that's going to indicate that there's gonna be a horizontal shift of the graph. Here, X minus three is in the standard form of X minus H and three takes up the spot of H. So this is going to be when H is equal to three. And since H is equal to positive three, that tells us that there's going to be horizontal shift of three units to the right of the graph. And that's the only transformation we're going to need to go from F of X to H of X. So let's go ahead and apply that transformation. If we draw another axis here, we are shifting the graph of F of X three units. To the right. That means that the new starting point of H of X is going to be at three comma zero. And this transformation doesn't change the shape of F of X. So we're still going to have this stretched curve going off to the right. And this is going to be the graph of H of X. With that being said, the option that accurately represents this transformation is going to be D C for D. The graph of F of X has a graph whose starting point is at the origin and one whose graph is stretched along vertically and H of X has the same shape as F of X. But the new starting point for H of X is going to be at three comma zero because of the horizontal shift. So D accurately represents all the transformations that we figured out, which means that D is going to be the answer to this problem. So I hope this video helps you in understanding how to graph using transformations. And I'll go ahead and see you all in the next video.

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. g(x) = √(x+1)

Key Concepts

Square Root Function

Graph Transformations

Horizontal Shifts

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