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. h(x) = -x³

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the function G. Of X where G. Of X equals negative two X cubed. And we are to graph it in relation to the function F of X which equals X cubed. And applying transformations to get from a five X two G of X. So for the first step I would like to give us more space to do this. So scrolling down. And now because we're graphing, I am going to draw a rough sketch of a coordinate plane will have a vertical Y axis, horizontal X axis. They meet together the origin in the middle. Now if you recall exactly what X cubed looks like on a graph that's great. If not you can always choose some X values, plug them into the function. Figure out the corresponding Y values and then graph them. So I am going to choose to 10 negative one and negative two for the X values. Now plugging those in two X cubed. That will be instead of X cubed. It will be two cubed. Well two cubed is eight. So looking at the graph going from the origin, going to the right two units and then up eight units approximately here. Now putting in one that will be instead of X cubed. It will be one cubed. One cubed is one. So from the origin going to the right one unit and up one unit that's approximately here. Now plugging in zero for X. That will be zero cubed zero cubed. 20. That's right at the origin. Right now plugging in negative one for X. Negative one. Cute equals negative one. So from the origin we're going to the left one unit and down one unit approximately here and now substituting negative two. That's negative two. Cute. That equals negative eight. So from the origin going to the left two units and down eight units that's approximately here. So our F. Of X looks something like this label this F. Of X and it intercepts both X and Y axes at the origin at 00. Now for G of X we are to apply the functions transformations. So what is the same between F. Of X and G. Of X and what's different? Well they're both X cubed. The difference is that G. Of X is being multiplied by a negative two. Alright so what does that mean? Well there are actually two things happening here. The negative and the two when we are multiplying by a negative number two our function and it's not happening directly to the function. If it were directly to the function it would be in parentheses like a negative X cubed. But that's not what's happening. It's happening outside of the X. So it indicates a vertical shift and namely it is reflecting over the X axis. So this whole thing is going to flip over the X axis. Now what does this times to mean? Well because that's again not happening directly to the exits. Not in parentheses with the X also getting cubed. It happens after the cubing is done then that too because it's a number greater than one indicates our Excuse me. So because it's not directly to the X. It's a vertical transformation. And because the two is a number greater than one it is vertical stretching. So the two things happening are reflecting over the X. Axis and vertical stretching. So we can draw this everything flipping over And it's going to look something like this. It will still intercept at the origin at 00 because it hasn't gone left right up or down anywhere. So now let's take this back up to the top and see what matches. Alright We look at our answer choices and we're looking for an intercept at 00 for our G. Of X. This matches with answer choice B. 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. h(x) = -x³

Key Concepts

Cubic Functions

Graph Transformations

Reflection Across the X-Axis

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