Graph the functions in Exercises 37–56.

y = (...

Question

Graph the functions in Exercises 37–56.

y = (1/x²) − 1

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says for the given function, draw the graph. And we're given the function Y is equal to 3 divided by X2 + 5. Below the problem, we're given an empty graph on which to draw our function. So we need to draw the graph of the function Y is equal to 3 divided by X2 + 5, and we're going to do this using transformations. So, we're gonna start off by looking at a basic function, which is going to be F of X. Which is equal to 3 divided by X squared. Now, if we look at a function of Y, we see that we have a + 5 being added to the quantity of 3 divided by X2. So that means we're going to need to shift all of our values for Y for a function F of X up by 5. So we're going to actually calculate a couple points for our function F of X. Now, before we do that, we noticed a couple of properties about our function F of X. We see that we have Xquad in the denominator, and when our denominator is equal to 0, we have a vertical asymptote. And so that's going to occur when X is equal to 0. So we have a vertical asymptote at X equal to 0. Also, as X approaches plus or minus infinity, our function F of X approaches 0. So we have a horizontal asymptote at Y equal to 0. Now, we'll need to calculate a couple points. On which to base our plot for our function Y. So we'll start off with X equal to 1. So we'll look at F of 1, and this is going to be equal to 3, divided by 1 squared, which is equal to 3. And our function F of X is symmetric about the Y axis, it's an even function. So that means that F of minus 1 is equal to F1. Which is equal to 3 as well. The next point we'll look at is when X is equal to 2, so we'll look at F of 2, and this is going to be equal to 3 divided by 2 squad, which is equal to 3 divided by 4. And again by symmetry F of minus 2 is going to be equal to F of 2. Which is going to be equal to 3 divided by 4. And the next X value that we'll look at is gonna be X equal to 3, so we'll have F of 3. And that's going to be equal to 3 divided by 3 squared, which is going to be equal to 1 divided by 3. And again, by symmetry F of minus 3. is equal to F of 3. Which is equal to 1 divided by 3. So now, in order to draw a curve for Y, We'll just need to take all these values for F of X and add 5 to them to shift our graph up by 5 units. So we're gonna start off by drawing our vertical and horizontal asymptotes. So we'll have a vertical asymptote. At X equal to 0. And we're going to have a horizontal asymptote, not at Y equal to 0, we'll need to shift that up by 5 units. We will have a horizontal asymptote at Y equals to 5. So we've drawn those two asymptotes with dashed lines on the graph. So now we'll plot a couple of points. So, when X is equal to 1, F of 1 is equal to 3, so we'll need to add 5 to that. And so that means when X is equal to 1, Y is going to be equal to 8 on our graph, we'll plot point there, and when X is equal to -1, Y is also going to be equal to 8 by symmetry. The next point we'll look at is when X is equal to 2. F of 2 is equal to 3 divided by 4, but we'll need to add 5 to that, so that'll be equal to 5.75. So, we'll plot that point on the graph. And likewise, when X is equal to -2, Y is going to be equal to 5.75. And the next point we'll look at is when X is equal to 3. F of 3 was equal to 1 divided by 3. We'll need to add 5 to that to shift our graph up by 5 units. So, our Y value is going to be approximately equal to 5.33. We'll plot that point when X is equal to 3, and when X is equal to -3, Y is also equal to approximately 5.33. So, now we need to just connect these points. With a smooth curve. And being sure that our curve actually approaches. Are vertical and horizontal asptodes. And when we do so, we get a graph that Similar to the one that we see on the screen. Where we have a graph that is symmetric about the y axis, where a function is approaching the two asymptotes and actually connects smoothly with all of our points. So the graph that we have on the screen is the answer to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Graph the functions in Exercises 37–56.

y = (1/x²) − 1

Key Concepts

Graphing Functions

Asymptotes

Transformations of Functions

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