In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) =...

Question

In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) = (1/x^2) to graph each rational function. g(x) = 1/(x + 2)^2 - 1

Graph of g(x) = 1/(x + 2)^2 - 1 showing rational function behavior.

Accepted Answer

Welcome back. I am so glad you're here. We are given the function F of X equals one over X squared. It's given to us. Yay, that's wonderful. And now we have to graph G of X where G of X equals one over two X plus one squared. Using transformation of affects. All right. Well what's different between our F. Of X and our G of X? Well, we note that there's this two being multiplied by R. X. And we also have this plus one. Well what do each of those mean they're both happening directly to our X. That's inside the parentheses they're getting squared and so they are both indicating horizontal changes. This plus one indicates the horizontal shift because it's adding it's going to be a shift to the left And this being multiplied by two. Again it's a horizontal shift and because it's being multiplied by a number greater than one then it's going to indicate horizontal shrinking. So those are the two major shifts that are going to happen. This thing is going to move to the left a little bit and it's also going to shrink horizontally. So we need to look for a graph that's doing those two things we can get a little bit more specific by looking for our new y intercept. We see that there they had an ASM tote at Y equals zero for F of X. So when it shifts to the left a little bit we'll have a new Y intercept. So we can solve for that. We can solve for a Y intercept by plugging in zero. So into G. Of X. We're going to solve for G. Of zero. That'll be one over in the denominator. It'll be a two times zero plus one squared. Well one stays the same on the top. Two times 00. So we've got zero plus one squared. Well that's 1/0 plus one is one so one squared. Well that's 1/1. It's one. So when in G. Of X when X equals zero, Y equals one. So we'll have a Y intercept at +01. So let's look for those shifts to the left shrinks a little bit and a Y intercept at 01. Let's look at our four answer choices. So answer choice A. Has an intercept at zero to that doesn't work. B. Has intercept at 00. That doesn't work. C. Hasn't intercepted 01. That works. Now let's look at D to see if that's the same. Well that hasn't intercept at 01 as well. Now the other shifts that we were talking about is that it's going to move to the left which of these moved to the left? Well that one is answer choice C. So answer choice D. Sorry you do not work. You went the wrong way. You went to the right So answer choice C. Works well done. We'll catch you on the next one

In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) = (1/x^2) to graph each rational function. g(x) = 1/(x + 2)^2 - 1

Key Concepts

Rational Functions

Transformations of Functions

Asymptotic Behavior

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