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

Question

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. ∛(-x+2)

Accepted Answer

Hello, today we're going to be graphing G of X as a transformation of F of X. So I'm just gonna go ahead and give us some room to do this problem but are given F of X is the cube root of X and we're going to be graphene G of X which is the cube root of the quantity negative X plus three. So before we graph anything, let's just go ahead and simplify G of X first. The first thing we can do to simplify G of X is to go ahead and factor out a negative one from the quantity negative X plus three factory, not a negative one is going to give us the cube root of negative one times the quantity X minus three. And 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 the cube root of negative one times the cube root of X minus three. And the cube root of negative one is going to evaluate to give us one. So what we are left with for G of X is the negative Q route Of X -3. Okay. So now that we have G of X simplified, let's go ahead and take a look at what F of X is going to look like on an axis while F of X is given to us as the general form of a cubic function. So when we graph that on the axis, the shape of the graph is going to start at the origin zero comma zero. And it's going to be essentially this long s shape. So this is the typical graph for a cubic function of X. But what we want to go ahead and do is graph G of X using transformations of F of X. So let's go ahead and identify the transformations. We need to go from F X two G of X. While there's going to be two in total. The first one involves the quantity X minus three as X minus three is similar of the form X minus H. So when you have a quantity of X, you're going to essentially have a horizontal shift here H is equal to three, which indicates that we're going to have a horizontal shift of three units to the right. So that's going to be the first transformation to go from F of X to G of X. The second transformation involves the negative coefficient as the negative coefficient flips the graph along the X axis. So now let's go ahead and alter F of X using the transformations of G of X. So if we draw another axis underneath F of X, We're going to first use our horizontal shift of three units to the right. So now our origin is gonna move from 00 comma zero since we're shifting this graph three units to the right. So now the origins at 30, but the shape of the graph stays unchanged. So this is gonna be our new graph using the first transformation and we're gonna use the last transformation which is us flipping this graph with respect to the X axis. So if we flip this graph with respect to the X axis, the origin is still going to be at three comma zero. But now the shape of the graph is flipped with respect to the X axis. So it's going to look something like this and this is going to be G of X. And with that being said, the graph that accurately represents G of X in the given options here is going to be a. So the answer to this problem is 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 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. ∛(-x+2)

Key Concepts

Cube Root Function

Graph Transformations

Horizontal Shifts

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