Graph ƒ₁ and ƒ₂ together. Then describe how applying the absol...

Question

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)

x² |x|²

Accepted Answer

Welcome back, everyone. Consider the functions F1 of X equals X cubed and F2 X equals the absolute value of X cubed. Graph them together and describe how applying the absolute value function in F2 modifies the graph of F1. Here we have an empty graph that will put F1 and F2. Now, let's focus on the first part of this problem. Let's graph F1 and F2 on our uh graph of X with the X and Y axis. Sorry, OK. Then, starting with F1, OK. Then for F1, which is equal to X cubed, notice that since F1 of negative X would be equal to negative X cubed, which would just be negative X cubed or the negative value of F1 of X, OK, then this tells us that F1 is an odd function. And because it's odd, that means it's symmetric about the origin. Now, in order to, for us, that's good to keep in mind while we graph F1. And let's just try to get a few points that we can use to draw the graph of F1, OK? So let's consider some arbitrary values of X. For example, let's take when X E equals -2. Then F1 of negative 2. It's gonna be equal to -2 cubed, which is going to be equal to -8. So that means the point negative to -8 is on our graph. On the other hand, F1 of -1 equals -1Q, which is -1, which means -1, 1 is on our graph. Next, we know that F1 of 0 will be equal to 0. So 00 is on our graph. F1 of 1 is going to be equal to 1 cub, which is 1. So 11 is on our graph. And then uh let's take when X equals 2. Then F1 of 2 equals 8, so 28 is on our graph. Now, we have a list of points that we can plot. So now we're gonna have -2, -8, -1, 1, 00, 11. OK. Uh, then we're gonna have 28. It's 28, it's gonna be right here. And now that we have these points, we can go ahead and connect them through a smooth curve. Well, as smooth as we can get. OK, so we know that it's gonna be something like this. So this is the graph of F1 of X. Next, let's try to plot the graph of F2X, OK? And we know that F2X equals the absolute value of XQ. No. Let's consider some arbitrary values for F2, OK. Again, using probably the same set of values from negative 2 to 2, OK? Now here, For F2 of -2, that means it's going to be equal to the absolute value of -2 cubed. The absolute value of -2 is 2, and the 2 cubed equals 8. So this tells us that -28 is going to be on our graph. Likewise, for F1 -1, it's gonna be the absolute value of -1 cubed. That's gonna be 1 cubed which equals 1. So that means uh -11 is on our graph. On the other hand, OK, F20 is gonna be equal to the absolute value of 0Q, which is 0. So 00 is on our graph. F2 of 1 is gonna be 1 cub, which is 1. So 11 is on our graph, and on the other hand, F2 of 2 is gonna be equal to 8. OK. So 28 is gonna be on our graph. And again, since we have these points, we can go ahead and plot, OK? So now, if we go back to our graph, we know that we'll have 28, OK, -11. 0011 and 28 So now notice that our graph is gonna look something like this. Let me try to draw it as close as I can to our curve, it's gonna look something. So let me draw it here. It's gonna look something like this, right? It's basically gonna look something like that. So now we've graphed F1 and F2 together. Now, how does applying the absolute value function in F2 modify the graph of F1? Well, notice here, OK, that when X is greater than or equal to 0, F12 X is identical to that of F1 of X because the absolute value of a non-negative number is the number itself. On the other hand, when X is less than 0 to the left hand side of our graph, the graph of F2 of X reflects the portion of F1 of X that lies in X in the region where X is less than 0 above the X axis because the absolute value function negates the negative input, making it positive before applying the cube. So this is why the graph of F2X appears as a U shape, with both sides of the y axis exhibiting the cubic behavior found in the positive side of F1 of X, where in comparison, F1 of X is a cubic curve symmetric about the origin that has the characteristic S shape of a cubic function. OK. So now, this tells us then. That for our graph, the graph of F2FX reflects the portion of F1 of X that lies in the region where X is less than 0 above the X axis. That's how the absolute value function in F2 modifies the graph of F1. Thanks a lot for watching everyone. I hope this video helped.

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
x² |x|²

Key Concepts

Graphing Functions

Absolute Value Function

Transformation of Graphs

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