In Exercises 67–69, begin by graphing the absolute value function...

Question

In Exercises 67–69, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. r(x) = (1/2) |x + 2|

Graph of the absolute value function f(x) = |x| in orange on a grid.

Accepted Answer

Welcome back. I am so glad you're here. We are given the function F. Of X equals the absolute value of X. And we need to transform it into G. Of X where G of X equals two times the absolute value of four X plus three and absolute value plus three. Well there's a lot of transformations going on here. We've got this X times the four and plus the three inside the absolute value bars. And then outside it's getting multiplied by two and adding three. So there's a lot going on before we identify what each of those four things means. I want to zoom in on what's happening inside the absolute value bars for just a second and I want to rewrite this and I want to factor out the four and then we'll have X plus and three divided by four is 3/4 because this will more accurately show us exactly what is happening to the center of our absolute value function. Okay, so now still have four different things happening. What do they each mean? I'm going to start closest to the X. So we have X plus 3/4 when that's happening directly to the X. And It is a positive value. That means we're going to be moving to the left and we're going to be moving to the left, 3/4 of a unit. Now, the next closest thing that's happening to X. Is this multiply by four when that is happening to the X than it indicates it is happening, it is a horizontal shift and specifically when it is being multiplied and it is greater than one. It is going to be horizontal shrinking that is occurring Alright. The next thing that is happening Here, let's look at this plus three on the end When we add three to our X. But it's not happening directly to the X. Like that. Then this indicates that we are moving up because it is a positive three. And then the last thing that's happening here is this multiply by two because that multiply by two isn't happening directly to the exits on the outside of that absolute value bars that indicates that it's going to be a vertical shift. And specifically because we're multiplying by a number that is greater than one, it is going to be vertical stretching. So those are the four different transformations happening to get to our G of X function. So let's look at our graph on the left. We are going to shift to the left from our origin 3/4 of a unit. So that's just moving a little bit to the left and then looking next we are going to move up three. So our new origin is going to be right. There are new center of our G of X. And we're going to have horizontal shrinking and vertical stretching. So it's going to look a little bit more like this. This isn't going to be exact but it gives us an idea when we're looking at the different transformation options for our answer choices. And if you need to have exact you can plot a few points by choosing some X. Values to graph and seeing where your Y intercept would be for example. But for now let's just look for a center at negative 3/4 or negative 0.75 for X. Value and then three for our Y. Value. So let's look for that. Now let's look at our answer choices -0.75 and three. Alright that again negative 0.753. And now looking at answer choice A. They've got negative 0.75 but it's negative 0.75 and zero. So A. Doesn't work for B. They have negative 0.75 but they have negative 0.75. And negative three. That one doesn't work for C. They have a positive 0.75 that one doesn't work. And four D. They have negative 0.75 and they have positive three. So answer choice D. Is the correct answer here. Well done. We'll catch you on the next one

In Exercises 67–69, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. r(x) = (1/2) |x + 2|

Key Concepts

Absolute Value Function

Transformations of Functions

Graphing Techniques

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