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) = 2√(x+1)

Accepted Answer

Hello, today we're going to be Graphene 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 X. And we are told that H of X is equal to three times the square root of the quantity X plus four. So before we graph H of X, let's go ahead and get an idea of what F of X is going to look like here F of X is equal to the traditional graph of the square root function. And the traditional graph of the square function is going to be a graph whose starting point is at the origin 000 and is essentially a curve that stretches off to the right. So this is going to be the graph of F of X. Now let's go ahead and identify what transformations we need in order to go from F of X to H of X. So in total H of X is in its simplest form and there's going to be two transformations. The first transformation comes from the quantity X plus four as the quantity X plus four is of the shape X minus H whenever you have an X minus H quantity in a function that's going to indicate a horizontal shift. But here X plus four is not in the standard form of X minus H. So we're going to need to rewrite at one time, we can rewrite X plus four as X minus negative four. And this is going to be where H is equal to negative four because negative four takes up the spot of H. So since H is equal to negative four, this is going to indicate that there is a horizontal shift of four units to the left. And that's going to be our first transformation for the graph H of X. Now let's go ahead and identify the second transformation and that's going to involve the coefficient of three C here, the coefficient of three is going to be greater than one and that's going to cause the graph to have a vertical stretch. So we're going to be stretching the graph vertically with respect to the Y axis. So now let's go ahead and a player to transformations to the graph of F of X. So the first transformation is going to be our horizontal shift of four units to the left. So that means that the new starting point for the graph is no longer at the origin but is now going to be at negative four comma zero. And for this transformation, the shape of the graph is going to be unchanged. So we're still going to have a curve that stretches off to the right For the next transformation, we're going to be taking this graph and we're gonna be stretching it vertically. So the starting point is still going to be at -400. But now that we're applying a vertical stretch, the shape of the graph is going to be looking much taller now. So now this is going to be the graph of H of X. And with that being said, the option that most accurately represents this is going to be a and just to double check the graph of F of X is going to be the traditional square root graph where the starting point is going to be at the origin. And we have one point right here at one comma one and for H of X H of X is shifted over four units to the left where the new starting point is at negative four comma zero and it's stretched vertically. So it's going to be much taller than F of X. So since this accurately represents what we came up with earlier, the answer to this problem is going to be a. 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) = 2√(x+1)

Key Concepts

Square Root Function

Transformations of Functions

Graphing Techniques

Watch next