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 let's take a look at F of X before we start to graph G of X. So for F of X, what we are given is the cube root of X. And this is one of our traditional graphs for the cube root of X because the cube root of X has a graph whose origin point is at the origin. So at 00 and the description of this can be described as a sideways s so this is the traditional graph for the cube root of X. Now let's take a look at G of X for G of X were given the cube root of negative X minus four. And before we start to figure out what transformations we need, let's go ahead and simplify G of X a little bit. We can start by factoring out the negative one from negative X minus four. And that's going to give us the cube root of negative one times X plus four. Now we can go ahead and separate the inside as two separate square cube roots because we have a product of negative one and the quantity X plus four. So separating it is going to give us the cube root of negative one times the cube root of X plus four. And finally, the cube root of negative one evaluates the negative one. So what we now have is the negative cube root of X plus four. Now for this graph, there's two transformations in total. The first transformation transformation for G of X is going to involve the quantity X plus four as this quantity is roughly in the shape of X minus H. So this is going to indicate that we have a horizontal shift but X plus four is not in the standard form of X minus H. We can rewrite this quantity to be X minus negative four. And this is going to be when H is equal to negative four. Since age is equal to negative four, that means we're going to have a horizontal shift of four units to the left. And the second transformation is going to be the negative sign as the negative sign indicates that we flip the graph with respect to the X axis. So let's go ahead and redraw F of X with these respective transformations. So the base graph for F of X was the sideways snake whose origin is at the origin of the axis. With our first transformation, we're shifting the origin of this graph four units to the left. So now what we're going to have is this graph whose origin is now and negative four comma zero. But the shape of the graph stays unchanged. So we still have this sideways snake. And for last transformation, since we're going to use the negative sign, we're going to flip this graph with respect to the X axis. So just to draw one more axis, the origin of the graph is still going to be a negative four comma zero. But now, since we're flipping this graph with respect to the X axis, the shape of the graph is now going to look like this. And this is going to be the function G of X. And with that being said, the graph that accurately represents G of X is going to be D. So the answer to this problem is D. So I hope this video helps you in understanding how the 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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