In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2...

Question

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+2)^2

Accepted Answer

Hello everybody I have doing right today. Today, we're going to look at this question that states graph the given rational function using the transformation of F of X is equal to one divided by X squared where the given function is G of X is equal to one divided by open parentheses, X plus three, close parentheses square. So the denominator is squared. Now the S provided are all sets of different graphs of A of X and GEO X with different points. So answer choice A has a point at F of X F of X has a point at one comma one and G of X has a point at four comma one answer choice B alpha X has a point at one comma one and G of X has a point at negative two comma one with answer choice C F of X has a point at one comma one and G of X has a point at one comma four. And with reso D G of X has a point at one comma negative two and F of X has a point at one comma one. Now they specifically tell us that they want us to use the transformation of our Afro X function. So to do that, we should know different transformation rules. So let's do a recall of set rules and being specific with this scenario. So let's recall that the graph of Y equals F of X plus C is the same as the graph of Y equals F of X but shifted left C unit. So whatever that C is and within the Y equals F of X plus C, that C is a constant. So that is the, that graph will be the same as the graph of Y equals F of X but shifted left C units. So in our case, all right, our case F of X is equal to one divided by X squared and G of X is equal to one divided by open parentheses, X plus three, close parentheses squared. Then this means that G of X is equal to F of X plus three because they're the same function. But we are including a plus three within with our. So we will shift F of X graph three units that and that is how we use the transformation of our alphabet function. Now, with this information, all we have to do is look at the different graphs provided and look at the different points provided and see which one has points, moving and shifting three units to the left. So we'll look at answer choice. A well, first we can eliminate answer choices. C and D because the graphs are shifting up and down. So the, we know those will deal with the Y axis and we're dealing with the X axis here, we're moving left and right. So we're moving horizontally so we can eliminate of C and D right away. And then we would be left with a choice A and choice B. So now we just have to look at the points and see which one has a G of X three units to the left of the F of X. So let's look at a choice A and we see that they have a point at one comma one, alpha X has a point at one comma one and G of X has a point at four comma one. Now to go from an X of one to an X of four, we are moving to the right. If our alpha X has A X at one and our G of X has an AX at four, they move it three units to the right and we want to move to the left. So answer choice A can be eliminated and we're left with an answer choice C, I mean, B we're left to have an answer choice B and we can just confirm by looking at the points once again. So F of X has a point at one comma one. Now, if we were to go from an X of one positive one, in this case, we shifted three units to the left, we would be at an X of negative two. And that's the other point they have, they have a G of X has a point at negative two comma one. So the F of X function has been shifted three units to the left. And that's exactly what we need. So just like that, we have figured out the graphs of our functions. So I hope that was helpful. And until next time.

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+2)^2

Key Concepts

Rational Functions

Transformations of Functions

Asymptotes

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