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) = (x − 3)^3

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the function G. Of X equals X plus two in parentheses, cute given the function F of X equal to X cubed and then applying transformations. So the first step I'd like to take is to get a little bit more room here. So we'll have the G. Of X. And our F. Of X. We are going to graph them both. So I'm going to start by drawing a coordinate plane. So we'll have a vertical Y axis and a horizontal X axis. They meet together at the origin in the middle. If you recall exactly what X cubed looks like on the graph, that's great. If not you can always choose a few X values, plug them into the function, find the corresponding Y values and then plot those points and that's where we'll start. I'm going to choose to 10 -1 and -2 for the X values. So now plugging two in for X. In the function that will be instead of X cubed two cubed two cubed is eight. So starting at the Origin will go to the right two units for X value and up eight units for our Y value. So approximately there now plugging one in for X cubed that will be one cubed and one cubed is one. So from the origin we're going to the right one unit and up one unit. So approximately there now plugging in zero for X zero cubed. That's zero. So that's right at the Origin. Now plugging in negative one for X negative one cubed equals negative one. So from the origin we're going to the left one and down one unit. Now for negative two. Plugging that in negative two cute equals negative eight. So from the origin going to the left two units and down eight units. So our F. Of X looks approximately like this label. This R. F. Of X. And it has X. And Y intercepts at the origin at 00. Now let's take a look at our G. Of X and see what transformations are occurring. Well first of all where is our affects within our G. Of X. What's similar? Well they're both X cubed. But the difference is that this G. Of X. The X. Is having to added to it before it can get cubed because that too is being added directly to the X. Directly to the function before it gets cubed. It's inside parentheses with the X. It indicates a horizontal shift. And because we are adding to it indicates a horizontal shift to the left. two units. So instead of passing through the origin we are now going to have an x intercept at -20 because we have shifted two units to the left. But that is the only transformation. So the rest of it. Well look just like our affects. I'll label this G of X. And now let's bring this up and see if we can find a corresponding graph from our answer choices. So again we are looking for an X intercept at negative two zero. We look at our answer choices and this matches with answer choice D. 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) = (x − 3)^3

Key Concepts

Cubic Functions

Graph Transformations

Horizontal Shifts

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