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)

4 - x² |4 - x²|

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 minus X2 + 5, and F2 of X, which is equal to the absolute value of the quantity of minus X2 + 5. Graph them together and describe how applied the absolute value function in F2 modifies the graph of F1. Below the problem we're given an empty graph on which to plot our functions. Now, the first part of this question wants us to graph our two functions together. So we'll need to calculate a couple of points for each of the functions in order to do that. So, we're going to start off with F1. And we're gonna start off by looking at the point when X is equal to 3. So, we're gonna need to calculate F1 of 3. And that's going to be equal to minus 32 + 5. And evaluating this expression, this is going to be equal to -4. The next point we'll look at is when X is equal to 2, so we'll have F1 of 2, and this is going to be equal to minus 22 + 5, and this is going to be equal to 1. And if we look at the next point, which is going to be when X is equal to 1, we'll have F1 of 1, and that will be equal to minus 12 + 5, this is going to be equal to 4. And next point we'll look at is when X is equal to 0, so we'll have F1 of 0, which is going to be equal to minus 02 + 5, and that's going to be equal to 5. Now for the remaining points, we can actually use the fact that our function F1 is an even function. So when we look at F1 of -1, this is going to be equal to F1 of 1, since we have an even function. And so this is going to be equal to 4. Likewise, if we look at F1 of -2, this is going to be equal to F1 of 2. Which is going to be equal to 1. And then finally, if we look at F1 of minus 3, That's gonna be equal to F1 of 3. Which is equal to -4. So we're gonna plot these points on our graph. So the first point we'll look at is when X is equal to 3, and we will have a Y value of -4. The next value that we'll look at is when X is equal to 2, we will have a Y value of 1. The next value is going to be when X is equal to 1, we'll have a Y value of 4. The next value is when X is equal to 0 will have a Y value of 5. When X is equal to -1, will have a Y value of 4. When X is equal to -2, we'll have a Y value of 1, and when X is equal to -3, we'll have a Y value of -4. So, what we're going to do now is just connect those points with a smooth curve. And what we will actually get is a problem. So that is going to be Increasing For negative values of X and it's going to reach a maximum. At X equal to 0, and then our function will decrease. Afterwards. So we're just going to connect all these points. With a smooth curve To get our parabola. So now what we need to do is plot our second function, F2 of X. And if we look at F2 of X, we see that F2 of X, It's just equal to the absolute value of F1 of X. So, this will allow us to quickly calculate our values for F2 of X, and we're going to choose the same values of X that we did for F1. So for F2 of 3. This is going to be equal to the absolute value of -4, which is just going to be equal to 4. Next, we'll look at F2 of 2, which is going to be equal to the absolute value of 1. And that's just going to be equal to 1. The next value that we'll look at is F2 of 1, which is going to be equal to the absolute value of 4, which is equal to 4. The next value is going to be F2 of 0, which is going to be equal to the absolute value of 5, which is equal to. 5 Continuing on to our negative values of X, we'll have F2. Of -1. Which is equal to the absolute value of 4, which is equal to 4. We'll have F2 of -2. Which is equal to the absolute value of 1, which is equal to 1. And then finally we'll have F2 of minus 3. Which is equal to the absolute value of minus 4, which is equal to 4. Now, at some point, our function F1 went from positive values of Y to negative values. So we actually want to find our root for our function F2, and that's going to occur when the quantity of minus X2 + 5 is equal to 0. So we can solve this for X by beginning by adding X2 to both sides of the equation. So we'll have 5 is equal to X2. Then taking the square root of both sides, we'll have X is equal to plus or minus the square root of 5, which is approximately equal to plus or minus 2.24. So now we're going to do is plot these points on our graph. So, starting off with X equal to 3. When X is equal to 3, or Y value is going to be equal to 4 for F2. Next, we're going to plot our root, which is going to occur when X is equal to approximately 2.24. The next point that we're gonna plot is when X is equal to 2, and that has a Y value of 1. When X is equal to 1 or Y value for F2 is going to be 4. When X is equal to 0, a Y value is going to be 5, when X is equal to -1, our Y value is going to be 4. When X is equal to -2, or Y value is going to be 1. Then we're gonna plot our second route, which is going to occur when X is equal to -2.24. And then finally, our last point that we're going to plot is when X is equal to -3, and we're going to have a Y value of 4. 4 F2. So now we want to do is actually connect all these points. With a smooth curve. So doing so, What we're going to see is that When F1 is positive, the two curves line up. So, we're just going to connect all the points with a smooth curve. And we get somewhat of a W shape. And In drawing this, what we can see is that when F1 is negative, F2 basically reflects the values of F1 about the X axis. So, it just basically makes the negative values of F1 positive. 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)
4 - x² |4 - x²|

Key Concepts

Graphing Functions

Absolute Value Function

Effects of Transformations on Graphs

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