Graph each rational function. ƒ(x)=4/(x-1)

Question

Accepted Answer

Mhm Hey, everyone in this problem, we're asked to draw the graph for the function given below the function we're given is F FX equals six divided by X minus three. We're given four answer choices, ABC and D, all four of them show graphs. Option A and B show vertical asom totes around X equals three. A horizontal Asymptote at Y equals zero. And they differ in the direction that the function is approaching that vertical Asymptote from option C and D show vertical asymptotes at X equals negative three horizontal as to its at Y equals zero. And again, they differ in which direction they approach that vertical Asymptote from. Let's start by writing out the function we were given. So we have that FX is equal to six divided by X minus three. We're gonna start with looking at X intercepts and Y intercepts. And what are the intercepts of this graph that helps us just determine a couple of points. So our X intercept is going to occur when Y is equal to zero. So we set F of X equal to zero, we get that zero is equal to six divided by X minus three there are no X values, we can plug into this function to give us zero on the right hand side. And so we actually have no X intercepts. Let's move to Y intercepts. These are gonna occur when X is equal to zero. So we sub X is equal to zero into our function. Can we get that F at zero is equal to six divided by zero minus three. So we have six divided by negative three, which is equal to negative two. And so we have the 0.0, negative two. Where are Y intercept is what about the horizontal as to? Hm Well, the degree of the numerator of this function is zero. OK? Because it's just a constant. So the highest exponent is just zero in the denominator, we have X minus three. That's a linear function. The degree is one. OK? The highest exponent on an X term is one. So the degree of the denominator is greater than the degree of the numerator. This tells us that we have a horizontal Asymptote at Y is equal to zero. So we have our X intercept or Y intercept or horizontal asy, what about our vertical as to while the vertical ASOM show is going to occur where the denominator of the function is equal to zero. So the denominator we have is X minus three. We set this equal to zero and we get that X is equal to three. OK? So we have no X intercept, we have a Y intercept and zero comma negative two, we have a horizontal Asymptote at Y equals zero and a vertical Asymptote at Y equals three. Now, with this vertical Asymptote, we can already eliminate answer choices C and T because they have the Asymptote, the vertical ascent tote at negative three. OK. Not positive three. So we can already eliminate those two answer choice. What we need to figure out is what direction the function approaches the vertical ascent to from in order to distinguish whether we have answer choice A or B. Now let's think about as X approaches the three and we're gonna go from the left first. Now, if X is approaching three from the left, our X value is going to be slightly less than three. OK. Numbers on the left of an X value on the graph are less that that means that X minus three is going to be negative. And so when we take our function six divided by X minus three, we have six divided by a negative number which is gonna give us a negative number. So our function F FX is going to be negative or less than zero. Now, if we approach from the right. Mhm mhm OK. Our X value is gonna be greater than three. OK. It's gonna be slightly larger as we get closer and closer to three from the right hand side. So X minus three is going to be bigger than zero six divided by a positive number, gives it a positive number. And so F of X is going to be greater than zero. So if we draw a sketch of this function, we found that we had a vertical as to at X equals three, we can draw that as a horizontal dash line at X equals three, we have a Y or a horizon, sorry, a horizontal as tode at Y equals zero. Do we draw that as a horizontal line at Y equals zero, we have no X intercept. So this function is not going across the X axis, we have a Y intercept at negative two. Mhm And now let's think about the direction that we're approaching this fun this vertical acid to we know that as we approach this vertical ascent tode our function is going either to positive or negative infinity because the denominator is getting closer and closer to zero. Now, from the left, we found that our function is negative and so we must be going to negative infinity. And so from our um Y intercept negative 20, the function is gonna continue downwards as it approaches that vertical asom tool, we know that as X gets really large, either positively or negatively, it's approaching the horizontal Asymptote. And so it's gonna approach the horizontal Asymptote as we go to the left, on the right side of this vertical ascent to our function is positive. So it's gonna be approaching positive infinity. And we know that as X gets really big, it's going to approach the horizontal ascent tode at Y equals zero. And so the function that we're graphing is going to look like the graph in option A, if we go back to our answer choices, and we look at that graph in option A, we have no X intercept A Y intercept at negative two, a vertical Asymptote at X equals three A horizontal ascent to at Y equals zero. As we approach the vertical Asymptote from the left, the function goes to negative infinity. And as we approach it from the right, the function goes to positive infinity, which is exactly what we found. Thanks everyone for watching. I hope this video helped see you in the next one.

Graph each rational function. ƒ(x)=4/(x-1)

Key Concepts

Rational Functions

Asymptotes

Graphing Techniques

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