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. r(x) = (x − 2)³ +1

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph G of X which is equal to parentheses X plus one cubed minus five. And we're to graph that in relation to the function F of X where F of X equals X cubed minus seven. So the first step I'm going to take is give us a little bit more room. And now since we'll be graphing, I'm going to draw a rough sketch of a coordinate plane with a vertical Y axis and a horizontal X axis they meet together at the origin in the middle. Now for graphing our F of XX cubed minus seven, we can choose a few X values, plug those in for X in our function, find the corresponding Y values and then plot those points. So I'm going to choose 10 and negative one for the X values. So for X cubed, that will now be one cubed minus seven while one cued equals one. So one minus seven equals negative six. So looking at the origin, we're going to move to the right one unit for X and then down six units for our Y value. So approximately there. Now let's plug zero in for X, that'll be zero cubed minus 70 cubed is zero. So zero minus seven equals negative seven. So we have a Y intercept at zero negative seven. So starting at the origin going down seven units, that's approximately here. I'll label that zero negative seven right now, let's plug negative one in. We have negative one cubed minus seven. Well, negative one Q is negative one negative one minus seven equals negative eight. So starting at the origin, we're going to the left one unit for our X value and then down eight units for our Y value. And now this looks like it could be a straight line. But if you recall from previous lessons, X cube does some curvy stuff and I mean, this is not going to be perfect but kind of this ish is our F of X. All right. Now, for our G of X, we want to graph that as a transformation of F of X. So let's see what's similar and where our transformations lie. Well, both G of X and F of X have the X being cubed but this G of X has a plus one inside the parentheses, getting cubed along with that X and then F FX had a minus seven and G FX only has a minus five. So let's break that down. What are these transformation transformations? Well, as we recall from previous lessons, we're going to start with the order of operations for transformations, which means we're going to look at what's happening directly to that X what's inside the parentheses getting cubed. And that's that plus one because it's happening directly to the X, that's a horizontal shift. And in particular, because we are adding one, it is going to be a horizontal shift to the left one unit. So we're going to be heading to the left one unit. All right. Now, how about this? Minus five, well minus five because it's not happening directly to the X. It's not inside the parentheses there. That's a vertical shift. And because we're subtracting five, it's a vertical shift down five units. However, take note that that ax was already shifted from the origin down seven units. That's where this minus seven really came into play. It shifted everything down seven units. If we had just taken zero and cubed it, then we would have had an intercept at the origin, but it got shifted down seven units. Well, this G of X is only shifted down five units. So let's start at the origin and go to the left one unit and then down five units. So we've got a center right here at negative one, negative five. And now we can draw that kind of curvy stuff because otherwise that would be the same. Now to get very particular, we could figure out what the Y intercept is of our G of X and our Y intercept is always going to have an X value of zero. So we can plug zero in directly to G of X function for X. So G of zero equals parentheses zero plus one cubed minus five zero plus one equals one. So we have one cubed minus 51 cubed is one, we have one minus five which equals negative four. So where X equals zero, Y equals negative four or in other words, we have a Y intercept at zero, negative four. All right. Now that we've got this put onto the graph, we can grab this and bring it up to the top and compare it with our answer choices. All right. So again, we're looking for a G of X that has a Y intercept at zero negative four, we look at our answer choices and this matches with 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. r(x) = (x − 2)³ +1

Key Concepts

Cubic Functions

Graph Transformations

Function Notation and Evaluation

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