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)

x³ |x³|

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says consider the functions F1 of X, which is equal to 1/2 multiplied by X cubed, and F2 of X, which is equal to the absolute value of the quantity of 1/2 multiplied by X cubed. Graph them together and describe how applying the absolute value function in F2 modifies the graph of F1. Below the problem, we're given an empty graph over which we'll plot our function. Now, in order to make this graph, we need to determine a couple of points to create our graph for F1 and F2. So, we'll start with F1, and we noticed that F1 is an odd function. So, we can use that symmetry to help us um reduce our calculations. So we're gonna choose some arbitrary points. So we're gonna start off by calculating F1, when X is equal to 2. So, we'll calculate F1 of 2. And that's going to be equal to 1/2 multiplied by 2 cubed, and when we evaluate this expression, this is equal to 4. The next point we'll look at is when X is equal to 1, so we'll calculate F1 of 1, and that's going to be equal to 12, multiplied by 1 cubed. Which is equal to 1/2. Next point we'll look at is when X is equal to 0, so we'll calculate F1 of 0, and that's going to be equal to 1/2 multiplied by 0 cubed, which is equal to 0. And now that we can use the fact that we have odd symmetry, meaning that it's symmetric about the origin, to calculate our last two points, when X is equal to -1 and X is equal to -2. And so, for F1 of -1, since we have odd symmetry, this is going to be minus F1 of 1, which is going to be equal to -12. And for F1 of -2, this is going to be equal to minus F1 of 2, which is equal to -4. So we're gonna go ahead and plot this points on a graph. So, starting off with X equal to 2, we see that our Y value for this point is going to be equal to 4. When X is equal to 1, our Y value on the graph is going to be equal to 1/2. When X is equal to 0, our Y value is going to be 0. When X is equal to -1, our Y value is going to be -12. And when X is equal to -2, we see that our Y value is going to be -4. So now we're going to just connect those points together with a smooth curve. And what we're going to get is the classic S shape for our cubic function. And after connecting all those points, we'll have our graph. So now we need to graph our function F2. And we noticed that F2. Of X is just equal to the absolute value of F1 of X. So, we're gonna use that fact to quickly calculate 5 values for F2, and we're gonna use the same X values that we did before. So, we'll have F2 of 2. It's going to be equal to the absolute value of 4, which is equal to 4. F2 of 1 is going to be equal to the absolute value of 12, which is equal to 1/2. The next value that we're gonna look at is when X is equal to 0, so we'll have F2 of 0, and that's gonna be equal to the absolute value of 0, which is equal to 0. And then for the last two points, we'll have F2 of -1, that's going to be equal to the absolute value of -12. Which is going to be equal to 12. And then we'll have F2 of -2. It's going to be equal to the absolute value of -4, which is equal to 4. So we'll plot these points on our graph. So doing so, the first two points, when X is equal to 2 and X equal to 1, are gonna be the same values that we had for our original function F1, and likewise for when X is equal to 0. However, when we look at F2 of -1, we get a value of 1/2, and when we look at F2 of minus 2, we get a value of 4. And so now we're going to do is connect these points. With a smooth curve. To create our function or graph of our function F2. So what we see is that For the curve that is greater than an equal to 0, when X is greater than or equal to 0, that the two curves overlap. So F1 and F2 overlap on our graph. But for values of X that are less than 0, we see that F2 basically reflects the values of F1 into the positive region. So above the X axis. And so that is going to be the answer to this question. So I hope that this explanation has been useful, and I'll see you in the next video.

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

ƒ₁(x) ƒ₂(x)
x³ |x³|

Key Concepts

Graphing Functions

Absolute Value Function

Effect of Transformations on Graphs

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