In Exercises 107-118, begin by graphing the cube root function, f...

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In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x+2) - 2

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Hello. Today we're going to be graphing H of X as a transformation of F of X. So what we are given is the function F of X is equal to one half times the cube root of four X. And the function of H of X given to us is going to be one half times the cube root of four x -64 -1. So before we start to graph H of X, let's go ahead and first figure out the shape of F of X. We can go ahead and first start by simplifying F of X because we have a product inside of the cube root. So we can separate four X as a product of two cube roots. And that's going to give us one half times the cube root of four times the cube root of X. Now we can go ahead and combine one half and the cube root of four together to give us the cube root of 4/2 times the cube root of X. Now, since F of X involves the cube root of X, we're gonna go ahead and graph F of X by first starting with the original function for the cube root of X, see the original function of the cube root of X has a has a graph whose starting point is at the origin zero comma zero. And the graph has this snake like shape to it. So this is going to be the original graph for the cube root of X. But in order to go from the original graph to F of X, we're going to need one transformation and that's going to involve the coefficient of the cube root of four over to see whenever we have a coefficient, we need to check the value because if the coefficient is less than one, it shrinks the graph vertically. And if it's greater than one, it stretches the graph vertically here, the cube root of 4/2 is going to produce a value that is less than one. So in order to go from the original cubic graph to F of X, we're going to need to shrink the graph vertically. So if we shrink Y equals to the cube root of X, then what we have is the graph of F of X equal to the cube root of 4/2 times the cube root of X. And this is going to give us a graph whose starting point is still at the origin 00. But now the graph is just shrunken vertically. So it's gonna be a little closer to the X axis. So this is going to be the graph of F of X. Now let's go ahead and see what transformations we need to go from F of X two H of X. But we're going to need to first simplify H of X C for H of X, we can first factor out of four from four X minus 64. If we do that, we have one half times the cube root of four times X -16 -1. And again, since we have a product inside the cube root, we can separate this as a product of two cube roots, this is going to give us one half times the cube root of four times the cube root of X minus 16 minus one. And if we combine the constants together, this is going to give us the cube root of 4/2. Okay. So now H of X is simplified, there are two major differences between F of X and H of X. First in H of X, we have the quantity X minus 16 inside of the cube root instead of just X. And we also have this minus one at the end of the graph that F of X doesn't have. So in order to go from, but we do have the same coefficient. So in order to go from F of X two H of X, we need two transformations. The first transformation is going to involve the quantity X minus As this is of the form X -H which indicates a horizontal shift. Now, here H is equal to 16. So we're going to have a horizontal shift of 16 units to the right. So that's gonna be our first transformation and our second transformation is going to evolve the negative one at the end of the, of the function C. Here, we can represent this as minus K at the end of the function. And whenever you have a K, a term that's going to indicate a vertical shift here, since K is going to equal to negative one, we're gonna have a vertical shift Of one unit down. So if we apply these two transformations to F of X, what we end up with is this graph H of X Where the origin is going to move 16 units to the right and one unit down. So the origin is going to be here At 16 Common -1. And well, actually, that's gonna be the only transformations, the shape of the graph is gonna stay unchanged from F of X, but it's still gonna be slightly shrunken. So this is going to be the rough graph of H of X. Actually let me go ahead and fix that a little bit slightly right here. And this is also where we're going to have an X intercept at 18 comma zero. And this is going to be the graph of H of X. Now, in the options above both of these graphs are plotted on the same axis. So the graph who has both of the option that has both of the graphs on the same axis that accurately represents what we came up with is going to be a. So A 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 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x+2) - 2

Key Concepts

Cube Root Function

Transformations of Functions

Graphing Techniques

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