Graph each rational function. See Examples 5–9. ƒ(x)=(x+1)/(x-4)

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked for the following rational function to graph our rational function is F of X equals in the numerator X plus two divided by the denominator X minus seven. And for our answer choices, we are given four graphs, we'll go into more detail comparing these graphs in a little bit. But for now, let's begin by building our own. We'll move this to a part of the screen where there's a little bit more space. And the first step I like to take when graphing a rational function is to see if this function can be further factored and X plus two and X minus seven, that's as factored as this one gets. So now we can draw a rough sketch of a coordinate plane. We'll have a vertical Y axis, a horizontal X axis. They'll come together at the origin in the middle and step. One of the graphing process is to identify any vertical asymptotes. And we do that by setting our denominator equal to zero. So X minus seven equals zero, we'll solve for X by adding seven to both sides and we'll have a vertical Asymptote at XX equals seven from the origin. We go to the right seven units and draw our vertical asom tote and label that X equals seven. And we got step one done. Now, step two, we'll be checking for any horizontal or oblique as some totes and for horizontal and oblique asom totes. We're going to be comparing the degrees of our numerator and denominator. And in this case, they both have an unwritten degree of one. They're both X to the first power. And that's the highest degree there. When the degrees of our numerator and denominator are equal to one another two things happen, we know that this is a horizontal Asymptote. And the second thing, the equation for that horizontal Asymptote is Y equals the leading coefficient of the numerator divided by the leading coefficient of the denominator. In this case, the leading coefficient of the numerator is one and the leading coefficient of the denominator is also 11 divided by one is one. So we have a horizontal asom tote at Y equals one from the origin. We go up one unit, we draw our horizontal asom tote and label that Y equals one. All right, step two is done. Step three, step three is to look for a Y intercept. We do that by solving four F of zero or setting our X equal to zero in the numerator instead of X plus two, we'll have zero plus two. It's all divided by in the denominator instead of X minus seven, it will be zero minus seven. Simplifying zero plus two is 20 minus seven is negative seven. So this is a Y intercept of negative two sevens or in other words, when X is zero Y is negative 2/ plotting that point from the origin will go down 2/7 of the unit. We can label that zero negative 2/7. Alright, that's step three. Now step four, step four is to look for X intercepts or an X intercept. We find that by setting the numerator equal to zero, our numerator is X plus two set that equal to zero, subtract two from both sides to solve for X. And we get an X value of negative two which means that when X is negative two Y is zero. So from the origin, we'll go to the left two units and we can label that negative two zero. Now, for step five, step five, we are going to ask if our line, our graph will intersect with the, in this case, we have a horizontal add some tote instead of oblique. So will our graph intersect with the horizontal Asymptote? We find that by setting our function which is F of X equal to the Y value of the Y intercept, which is usually shown as B as in like Y equals M X plus B. And so this time our F of X, our function is X plus two divided by X minus seven, we'll set that equal to the Y value of our Y intercept, which is negative 2/7. And to solve here, we can just cross multiply we have a fraction equal to a fraction. So taking the numerator from the left X plus two, multiplying that by the denominator on the right, which is seven setting that equal to the numerator on the right, negative two multiplied by the denominator on the left, which is the quantity of X minus seven. Now we can distribute seven multiplied by X plus two is seven X plus 14. And on the right negative two, distributing negative two multiplied by X minus seven is negative two, X plus 14, subtracting 14 from both sides, those fourteens are gone and we just have seven X equals a negative two X. And that makes no sense whatsoever. When this makes no sense whatsoever, then there is no intersection with the Asymptote. So here, no, we do not intersect with the horizontal asom tote. All right. Step six, step six is to plot some points. And we want to be strategic. We're going to create an X Y chart and we're going to choose X values that are between our intervals, the intervals are outlined by infinity, negative infinity, our X intercepts and the vertical asom totes. So we want our first X value between negative infinity and the X intercept at negative two. And for that, I'm choosing an X value of negative three. We want our next X value to be between the X intercept negative 20 and the vertical asom tote X equals seven. And for that, I'm choosing the X value of zero. We want our last X value to be between the vertical Asymptote, X equals positive seven and infinity. And for that, I'm choosing the X value of a positive eight. You don't have to choose the same ones. Now, what we do is plug those X values into our function and see what the corresponding Y values are. You're welcome to pause the video and figure that out. I've already done this. When I plug in negative three, I get a Y value of 1/10 plotting that point from the origin will go to the left three units and up 1/10 of a unit approximately there for the X value of zero. We already did this. This is the Y intercept, the Y value is negative 2/7 and that's already been plotted for the X value of eight plugging that in we get a Y value of from the origin. We go to the right eight units and up 10 units, something like here. Now, all we need to do is connect the dots. So we know that we're not going through the horizontal Asymptote at any point. So for our first line that we're drawing, that's going to be to the left of our vertical Asymptote X equals seven. And it's going to be below our horizontal Asymptote, Y equals one. The line heads toward negative infinity for the X values along but just underneath that vertical or excuse me, horizontal Asymptote Y equals one and it will go through our points negative 3, 1/10 through the X intercept, negative 20 through the Y intercept zero, negative 2/7. And then it will slowly keep going heading toward negative infinity for the Y values along the vertical Asymptote X equals seven. There's one of our lines now for the other line that's going to be to the right of our vertical Asymptote X equals seven. It's going to come down from positive infinity for the Y values go through our 70.8 10 hug up along the Asymptote of X equals seven and then it will turn and then it will head toward positive infinity for the X values along the horizontal Asymptote Y equals one. And there's our graph. Now we just need to see which answer choice most closely corresponds with it. So we'll grab this and the head up to a different part of the screen and we can start off looking at answer choices C and D. Sometimes it helps to start off just comparing asymptotes. We have a vertical Asymptote of X equals seven. We answer traces C and D have vertical asymptotes of X equals a negative two. So we can rule both of those out. Now, we can look at answer traces A and B, we have vertical asymptotes of X equals seven for both A and B. Great. Now let's look at the horizontal asom totes, we have a horizontal asom tode of Y equals one. Well, so does answer choice. A answer choice B has a horizontal asom tode of Y equals negative one that doesn't match, we can rule that out and we can just quickly check to the other parts of our graph with answer choice A, we've got an X intercept of negative 20. Great. So does answer choice A, we have a Y intercept of zero, negative two sevens. Great. So does answer choice A and it looks like all of the lines are drawn in the correct quadrants. So answer choice A works well done. We'll catch you on the next one.

Graph each rational function. See Examples 5–9. ƒ(x)=(x+1)/(x-4)

Key Concepts

Rational Functions

Asymptotes

Intercepts

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