Begin by graphing the cube root function, f(x) = ∛x. Then use ...

Question

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)

Accepted Answer

Hello, today we're gonna be graphing the function H of X as a transformation of F of X. So what we are given is F of X is equal to the cube root of 27 X and we are given H of X which is equal to one third Times The Cube Root of 27 X -27. So before we graph H of X, let's go ahead and get an idea of what the shape of F of X is going to be, we can do a little bit of simplification with F of X because currently there is a product inside the cube root of F of X 27 times X. So we can rewrite this as a product of two cube roots. This is going to give us the cube root of 27 times the cube root of X and the cube root of 27 is going to evaluate to give us three. So what we have is three times the cube root of X and this is going to be the function of F of X simplified now, F of X currently uses the cube root of X. So the parent function or the standard graph of the cube root of X is going to look like this, we're going to have a graph whose origin or starting point is at the origin zero comma zero. And this graph is going to have a snakelike feature which makes the graph looks something like this. Now, for F of X, the difference between Y and F of X here is that F of X has a coefficient of three. So in order to go from the standard cubic graph to F of X, we're going to need exactly one transformation and that involves the coefficient of three. See here, the coefficient of three is greater than one. And that means that this coefficient is going to result in a vertical stretch of the graph. So if we want to go ahead and graph Y of X, we're going to take the traditional graph whose origin is still or starting point is still at the origin. But we're going to stretch this graph vertically. So for F of X, which is equal to three times the cube root of X, we're not changing the origin at all, but we're going to stretch the graph vertically. So now the graph is gonna look something like this. So this is going to be the graph of F of X. Now let's see what transfer transformations we need in order to go from F of X two H of X. But we first need to simplify H of X. The first thing we can go ahead and do is factor out of 27 inside the cube root of H of X and factory. Now 27 is going to give us one third times the cube root of 27 times the quantity X minus one. And now that 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 third times the cube root of times the cube root of the quantity X minus one. And again, we know that the cube root of 27 evaluates to three. So we're going to have one third times three times the cube root of the quantity X minus one. And finally, one third times three is going to give us one. So what we are left with is the cube root of X minus one. All right. Now let's go ahead and list out the transformations. We need to go from F of X two H of X. The first major difference is that we don't have a coefficient of three in the function H of X. So the first transformation is going to be the coefficient of one which is going to result in a vertical shrink. Or in other words, it's going to return the original graph to its original form. And the second coefficient involves the quantity X minus one as X minus one. Is representative of the quantity X minus H. And since we have an X minus H quantity, this is going to result in a horizontal shift of the graph here in X minus 11 takes the space of H. So H is going to equal to one or in other words, we're going to have a horizontal shift of one unit to the right. So these are gonna be the two transformations we need in order to graph H of X. So starting with F of X, the first transformation is going to be a vertical shrink. Since we no longer have that coefficient of three, the starting point is still going to be at the origin zero comma zero. But now the graph is gonna look more reminiscent of the original cubic function. And finally, for the second transformation, we're going to need a horizontal shift one unit to the right. So a horizontal shift, one unit to the right is now going to put the starting point of the graph at 1:00. And this is still not going to change the shape after the first transformation. So the new graph is gonna look something like this and this is going to be the graph of H of X. Since it's the graph of H of X and F of X, the option that most accurately represents these two is going to be D. And let me explain why for the graph of F of X you can see that F of X is slightly stretched along vertically. And for H of X, the new starting point is going to be here at one comma zero and the graph is going to be vertically smaller compared to F of X. So D is going to be the answer for 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.

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)

Key Concepts

Cube Root Function

Transformations of Functions

Graphing Techniques

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