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) = (1/2)(x − 2)³ – 1

Accepted Answer

Welcome back. I am so glad you're here. We are given this function where F of X equals negative X cubed. And we are to use that function apply transformations. And then graph G of X which is equal to one third times parentheses. X plus two cubed plus five. That is a lot of transformations. Alright well what I'm going to do first is give us a little bit more room and next I'm going to draw a rough sketch of a coordinate plane 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 F of X equals negative X cubed looks like that's great if you don't you can create an Xy table, assign some X values. Plug those into the original function. Figure out the corresponding Y values and then graph those points and that's what I'm going to do in case you don't know exactly what that looks like. I'm going to choose X values of negative two negative 101 and two. Now filling those into the function we have a negative instead of X cubed. It's negative two cubed while negative two cubed equals negative eight. So a negative negative eight equals a positive eight. Now plugging in negative one negative negative one cubed negative one cubed equals negative one and negative negative one equals positive one. Now for zero we have negative zero cubed. Well zero cubed. That's zero and negative zero is zero. Now plugging in one. Negative one cubed one cubed equals one. So negative one equals negative one. Now plugging into we have a negative too cute. Two cubed is eight. So a negative eight equals negative eight. Now let's plot these points. We start at the origin 00 in the middle for our first point negative 28. We're going to go to the left two units for our X. Value of negative two. And we're going to go up eight units for our Y. Value of eight. So that's approximately here for our next point negative 11. We're going to start at the origin go to the left one unit for our X. Value of negative one and up one unit for our Y value of one. That's approximately here for the next 10.0. That is the origin. I'm going to mark that 00 because that is our X. And Y intercept. Now for the next 0.1 negative one will go to from the origin to the right one unit for our X. Value of one and down one unit for our Y. Value of negative one. That's approximately here. For the last 0.2 negative eight. Starting at 00 at the origin we're going to go to the right two units for our X value of two and down eight units for our Y value of negative eight. And that's approximately here. If we connect these dots we've got a function looking like this here is our affects. All right now we need to apply transformations in order to get from our affects to our G of X. So first, what's the same between our X and our G of X? Well not much. They're both an X cubed. But other than that there are quite a few differences. So let's follow the order of operations for transformations of functions. First we look for horizontal shifts and the horizontal shift here is this plus two that we have in parentheses with the X. It's happening directly to the X. It's also getting cubed. So that is a horizontal shift and because it is adding to it is going to go to the left two units for that horizontal shift. Okay, there's our first transformation. Next we're going to look for any stretching or shrinking. The stretching or shrinking is right here. This one third that's being multiplied. That's happening on the outside of the parentheses. It's not happening directly to the exits. Not also getting cubed. So that is a vertical transformation. And because that one third is between zero and one that indicates vertical shrinking. Okay, next we're going to look for any reflections. Well, the original F of X was reflected over the X axis. And so because our function G of X is not being multiplied by a negative, it's going to reflect in terms of what the F of X is doing. It's going to show the F of X reflected over the X axis. So in terms of the F of X. R. G of X is reflected. And lastly we look for vertical shifts. That's this plus five here on the end. That is not happening directly to the X. It's not getting cubed, it's not in the parentheses. So that is a vertical shift. And because it is adding five, it is a vertical shift up five units. Okay, so a lot of transformations happening here. So the horizontal shift to the left two units. So it's going to head this way. Two units. It's going to show vertical shrinking. That's kind of hard to demonstrate. But we can find some points that show the vertical shrinking shortly reflected over the X axis. So instead of looking this way it's going to reflect this way, but it's going to the left and it's shifting up five units. So it's going to be something like this ish that's not perfect. But centered around this point that is two to the left and up five that approximately is our G of X. Now, in order to make it easier to find the appropriate answer choice, we can find our X and Y intercepts even though it looks like I didn't really draw a Y intercept, there is one and in order to do that, I'm going to make an xy table. And because these numbers are pretty complex, I'm just going to find the intercepts, I'm going to put zero in for X and then zero in for Y. So starting with zero in for X. So this will be G of zero equals one third instead of X plus two. It will be zero plus two cubed plus five. Doing what's in parentheses? Zero plus two is two. So we have two cubed, that's still times one third and plus five two cubed is eight. So we have eight times one third plus five. Eight times one third is eight thirds plus five. Well five we can convert that to a fraction. So we'll still have eight thirds five with the common denominator of three. That'll be 15 3rd. Eight plus 15 is 23. So we have 23 3rd converting that to a decimal because that's the way our answer choices are that is 7.667. So we have a Y intercept if you imagine that it exists here ish that's going to be a Y intercept of 7.667. And I'll redraw this a little bit so that it kind of looks like that's what it's doing and there we go, modified G of X. Kind of and now let's plug in zero for our Y value and see what's going on. So we'll have zero equals one third X plus two cubed plus five. Well we can subtract five from both sides of the equation. We'll have negative five equals one third times X plus two cubed. We can multiply both sides of the equation by three to get rid of that one third. And then on the left will have negative 15 equals on the right will have X plus two cubed. We can take the cubed root of both sides of the equation. And so on the left we have the cubed root of negative 15. And on the right that equals X plus two. Now we can subtract two from both sides and we have negative two plus the cubed root of negative 15 equals X. You can put that into a calculator and that will equal a negative 4.466. So there's our X intercept at negative 4.4660. Okay. There is our graph and now we can drag this up top and see which answer choices match. Ish probably looks slightly prettier than my rough sketch. Okay. So specifically let's look for an affects that has X. And Y intercepts at 00 and we want a G. Of X. That has an X intercept at negative 4.4660. And a Y intercept at 7.667. And generally the G. Of X. Is shifted to the left and then up. Alright looking at our answer choices. Answer choice A. Has an F. Of X. With an X. And Y intercepted 00. Has a G of X. With an X intercept at negative 4.4660. And it has a Y intercept at 7.667 answer choice A. You work 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) = (1/2)(x − 2)³ – 1

Key Concepts

Cubic Functions

Transformations of Functions

Graphing Techniques

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