In Exercises 95-106, begin by graphing the standard cubic functio...

Question

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

Accepted Answer

Welcome back. I am so glad you're here. We are given the function F. Of X which equals X cubed plus one. Were to use that to graph G of X which equals X cubed minus two. So for the first step I'm going to give us some more room to work with. Be a mess for a second. Bring this down. So we'll start off by graphing half of X. And then once we've got that graft we'll apply transformations to graph G. Of X. I'll draw a rough sketch of a coordinate plane for our graph will have a vertical Y axis and a horizontal X axis. They come together the origin in the middle. Now to graph F. Of X if you know exactly what X cubed plus one looks like. That's wonderful. If not you can always draw an X. Y. Table, plug in some X values to the function. Come up with the corresponding Y values and then plot those points. So I'm going to choose to negative one and negative two for my X values. Now let's plug those into the function instead of execute will have two cute plus one and two cubed equals eight. So eight plus one equals nine. Now for one for our X value one cubed +11 cubed is one. So one plus one equals two. Now plugging zero in for our x value zero, cute equals zero. So zero plus one equals one. Now plugging negative one in for x value negative one cubed is negative one. So negative one plus one equals zero. Now plugging negative two in for X value, negative two cubed is negative eight plus one. Negative eight plus one is negative seven. Now let's plot these points. Starting at the origin in the middle 00. We're going to go to the right two units for our X. Value for 29. And we're going to go up nine units for our Y. Value. And that's approximately here. Now for the 0.12 we'll start at the origin we'll go to the right one unit for our X. Value and go up two units for our Y value approximately there. Now for 01 we will go nowhere for our X. Value and will go up one unit for our Y. Value. I will label this as 01 because that is our Y intercept. Now for negative 10. Starting at the origin will go to the left one unit for X. Value. And we'll go nowhere for our Y value. I'll label this because this is our X intercept at negative 10. And the last one here will start at the origin will go to the left two units for our negative two and then down seven units for our negative seven. Now connecting these will look something like this for our function F. Of X. Alright we've got F of X graft wonderful. Now we need to apply the transformations in order to graph G. Of X. Well first let's look at what's similar between them both. F of X and G of X. R X cubed. That's the general shape of this graph. Now our F of X shifted one unit for this plus one. This plus one is not directly connected to our X. It's not inside of the parentheses getting cubed alongside with the X. So because it's not tied directly to the X indicates a vertical shift. And specifically because it's adding one, it's a vertical shift up one unit. Now the difference between the F. Of X and the G of X is the G of X. Also has a vertical shift applied that minus two is not applied directly to the X. So it's a vertical shift. But this time because we're subtracting two, it's a vertical shift down two units. So looking at our graph, the F of X has been shifted up from the origin one unit. We see that Y intercept at 01. Well, R G of X needs to go down two units from the origin. So that's going to have a Y intercept at zero negative two. But otherwise, it will look identical to the F. Of X. Here's our G. Of X. Otherwise it will look identical because the X cubed and the X cubed are the same. All right. So, we've got this graft. Now let's see if we can bring this up. Might need to shrink it a little bit and then we'll look for the matching answer choice. Alright. That almost kind of works. Okay. So we've got for F. Of X we're looking for a Y intercept at and for our G of X we're looking for a Y intercept at zero negative two. Looking at our answer choices, answer choice A has an F. Of X. Y intercept at 01, and it has a G. Of X. Y intercept at zero negative two. None of the rest of the answer choices have those same intercepts. So the correct answer is answer choice A. Well done. We'll catch you on the next one.

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

Key Concepts

Cubic Functions

Graph Transformations

Vertical Shifts

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