In Exercises 95–98, use long division to rewrite the equation for...

Question

In Exercises 95–98, use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to rewrite the equation of F in the form quotient plus remainder divided by divisor using long division and graph the function using the equation of this function and the transformation of one divided by X. Our function is F of X equals the quantity of three X plus five divided by the quantity of X plus eight. Our answer choices are answer choice A F of X equals three plus 19 divided by the quantity of X minus eight. And then it has a graph answer choice B F of X equals three minus 19 divided by the quantity of X minus eight. And then there's a graph answer choice C F of X equals negative three minus 19 divided by the quantity of X plus eight. And then there's a graph and answer choice D F of X equals three minus 19 divided by the quantity of X plus eight. And then there is a graph. All right. So now we need to start off by doing the long division, we're going to take our numerator which is three X plus five. Divide it by our denominator which is X plus eight. Now, we recall from previous lessons on long division, we're going to ask ourselves how many times does X go into three X or in other words, three X divided by X? And that's a positive three. We put that above the five. They are like terms. Now we take the three and multiply it by both the X and the eight. So we'll have three multiplied by X which is three X and three multiplied by eight, which is a positive 24. Now, we are going to subtract three X plus five minus the quantity of three X plus 24 or in other words, we need to be sure we subtract both of those terms. Well, three X minus three X is nothing, there are no axes and five minus 24 is a negative 19. There is nothing else here to divide. So we have our answer and we can write it in the format we have quotient which is three plus the remainder in the numerator, which is negative 19 divided by the divisor in the denominator which is X plus eight or in other words, this is the same as three minus divided by X plus eight. All right, we are going to need this. This is the first part of our answer. But now let's work on doing that graphing. So we can draw a rough sketch of a coordinate plane. We have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. We are asked to graph considering the transformation of one divided by X. And now we recall from previous lessons, what the graph of one divided by X looks like. If you don't recall, you can always set up an X Y table plot some X values. You could choose negative five negative 101 and five plug those in for X and then determine the Y values. So if you put negative five, that would be negative 1/5 negative one, that would be negative 10, that would be undefined. One would be a positive one and five would be a positive 1/5. So that can always give you a clue. If you don't recall what it looks like as it stands, we're going to have our function be in the, the first quadrant and the third quadrant, it's come going to come down from negative or excuse me from positive infinity for the Y value. And it's going to pass through that 0.11 and then head toward positive infinity for the X axis. And it has an asom tote along the Y axis because that's where we would have, if we put zero in for X, it would be undefined. And then we're also in the third quadrant coming from negative infinity for our X axis coming along hitting the point negative one negative one. And then heading to down toward negative infinity for the Y axis. And there is our graph of one divided by X. Now, what does that have to do with our F of X? Well, our function is very close to one divided by X. If we take our three minus 19 divided by X plus eight, it's similar except it has that 19 in the numerator. But we can factor that 19 out in front, we all have three minus 19 multiplied by the quantity of one divided by X plus eight. And there we have our one divided by X in the parentheses. And now we can figure out what our transformations are. So we recall from previous lessons on transformations, we can draw another rough sketch of a coordinate plane. We'll still have a vertical Y axis and a horizontal X axis. And now we can sketch the transformations in the previous graph and then have our final graph down in this second coordinate plane. The first thing we look for our are uh horizontal shifts. So horizontal shifts are going to occur, happen directly to our X directly to our function. And that's in here, this X plus eight, the X is having eight added to it. And that means that we're going to have a horizontal shift to the left because that eight is positive, it's going to the left eight units. We can draw lines to let us know that it's going to the left eight units. The next thing we look for is any stretching or shrinking? Well, that X doesn't have anything being multiplied directly to it, but there is on the outside of the parentheses that 19 being multiplied. So that does indicate stretching or shrinking. So which one is it? Well, it's not happening directly to the X. So it's going to be vertical and the number, the 19 is greater than one instead of between zero and one. So that indicates stretching and it's vertical stretching by a factor of 19. So that's harder to note without specifics. But you can imagine sort of a vertical stretch by 19. And next, we are going to look for any reflections. So reflections are going to be multiplying by a negative. There aren't any happening directly to the X inside of parentheses, but there is this negative in front of the 19 that's being multiplied to our function. So we are going to have a reflection because it's not happening directly to the X. It's going to reflect over the X axis, which means this first quadrant is coming down to the fourth quadrant and this third quadrant is coming up to the second quadrant. All right. The last transformation that we look for are any vertical shifts and that would be any numbers being added or subtracted outside of our function. And here this three is being added. So that is a vertical shift. It's not happening directly to the X and it's a positive three. So it's going to go up three units. So this whole thing will go up three. OK. Now let's figure out how to show all of these transformations on our graph. Remember we had an ASOM tote along the Y axis because we can't have it where X equals zero. And so now it's shifting to the left eight units. Our asom tote is now going to be where X equals negative eight. And we have vertical stretching by a factor of 19. We can just imagine that it's stretched. And next, we have it reflecting over the X axis. So coming down into different quadrants and it's also shifting up three units. So it will look something like this. All right, similar to that. Not exactly but pretty close. All right. Now, let's look for that answer that we determined where we are F of X equals three minus 19 divided by X plus eight. Looking at answer choice A that's three plus 19 divided by X plus minus eight. That one is not what we're looking for. Did I do this wrong? No, it's X plus eight. OK. They had X minus eight. Got it. All right. Answer choice B F of X equals three minus 19 divided by and they again have X minus eight. That's not what we're looking for. All right now, let's go check out our C and D, we can bring this down. OK. Answer choice C F of X equals a negative three minus 19 divided by X plus eight. Well, they got the X plus eight but they have that negative three. That's not what we're looking for in answer choice. D three minus 19 divided by X plus eight. Ok, that's what we've got. Does our graph match while this one has shifted? It looks like it has that asom tote a long X equal negative eight. That looks good. They flipped over the X axis and they shifted up a little bit and stretched. Great answer. Choice. D is correct. Well done. We'll catch you on the next one.

In Exercises 95–98, use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

Key Concepts

Polynomial Long Division

Rational Functions

Transformations of Functions

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