Solve each problem. Work each of the following. Sketch the graph ...

Question

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Accepted Answer

Welcome back. I am so glad you're here. We are told the characteristics of a rational function F are given below. Draw the graph of F of X. We have X intercepts of 50 and 70. A vertical asom tote of X equals six and a horizontal asom tote of Y equals three. And we're told the graph does not intersect the horizontal asom tote. Well, there are a couple of different paths we can go down to solve this. So to draw the graph of F of X, we have a lot of information to start that we can draw a rough sketch of a coordinate plane. We have a horizontal X axis and a vertical Y axis. We have X intercepts at 50. So from the origin, we'll go to the right five units, we can label that 50 and at 70. So from the origin, we can go to the right seven units and label that 70, a vertical Asymptote at X equals six. So from the origin, we go to the right six units and draw a vertical asom tote label that X equals six. And we know we have a horizontal ASOM tote at Y equals three. So from the origin, we go up three units and we draw a horizontal asom tote and label that Y equals three. So we have some great information here and we could guess what's going on with our graph. However, we also have the tools and the information to know exactly where to plot our points for the graph. So we recall from previous lessons that X intercepts are found when we set the numerator equal to zero. So if we have X values of five and seven, that means we've got two factors in our numerator, one is where a will equal zero, but we know that X equals five. So to get X equal to zero, we subtract five from both sides and we have one factor is X minus five. The other factor in the numerator. Well, we know that we need to get that to zero. So we have X equal to seven to get that equal to zero, we subtract seven from both sides and then we have another factor of X minus seven. All right. So that's the useful information from the X intercepts. How do we derive the vertical asom tote? Will we find the vertical asom tote when our denominator is equal to zero? So in our rational function, we know that that's a function that is a fraction. So this is our F of X, it's equal to this fraction. So now setting our denominator equal to zero that we know we start from X equals six, but we want to get that to be zero. So we subtract six from both sides and we have a deno denominator with a factor of X minus six. So we're starting off now, we have more information. We're told that the horizontal Asymptote is Y equals three. Well, we get a horizontal Asymptote of a number like Y equals three when we have a coefficient from the numerator, a leading coefficient divided by a leading coefficient of the denominator. And that is true when the degree of the numerator is equal to the degree of the denominator. And that's when we can get a number like three. If we had different situations, like the degree of the numerator is greater or less than the denominator, we would be dealing with slant or oblique asymptotes or we would have our horizontal asom tode at Y equals zero. So because it's the number three, we know the numerator is equal to the denominator. And when we divide those coefficients, it's going to equal three. So we can put in our numerator, we can put a three and then nothing in the denominator will just be one for our leading coefficient. However, it says that our numerator's degree needs to be equal to our denominators degree. Right now, our numerator has a degree of two and the denominator has a degree of one. So how do we get them to be the same. They need to both be two because that's the larger number. How do we get the denominator's degree to be two while maintaining a vertical asom tote at X equals ze- X equals six. Excuse me? Well, we can just put another factor X minus six and that will give us our vertical asy tode of X equals six. That one's fine. But now we have a degree of two in the denominator. So now we have a function where we can plug in numbers, we can create an X Y table and we can plug in values for X figure out what Y is equal to and plot those points. So we're calling from previous lessons. We always want to choose from our intervals. So we want an X value from negative infinity to that first X intercept at 50. We want an X value for the um interval between 50 and X equals six. That vertical as to we want another X value between our vertical Asymptote X equals six and that intercept of and then one more X value between our X intercept of 70 and positive infinity. So I chose X values for the first interval. I chose the X value of zero. That'll also give us our Y intercept. I choose an, chose an X value of 5.5 for between 50 and six. I chose an X value of 6.5 between X equals six and 70 and then, or between 70 and infinity, I chose an X value of eight. You're welcome to pause the video, plug those X values into our function, find out what the corresponding Y values are. But here's what I got for X equal zero, Y is equal to 35 12th, plotting that point from the origin. We go up 35 12th of a unit which is less than three, just between two and three, just under three. For the X value of 5.5, I got a Y value of negative nine from the origin. We go to the right 5.5 units and then down nine units. So somewhere about here for the X value of 6.5, I also got a Y value of negative nine from the origin going to the right 6.5 units and then going down nine units approximately there. And for an X value of eight, I got a Y value of 9/4. So from the origin, going to the right eight units and then going up just over two units for 9/4. So now we can connect those points. So drawing our function or our graph, we are going to be below our horizontal Asymptote. We're told the graph doesn't intersect the horizontal Asymptote. So we'll be below Y equals three toward negative infinity for the X values. And then we'll come along and we'll hit our Y intercept of 35 12th. Then it's going to curve, it's going to hit our X intercept of 50. And then it's going to head down toward negative infinity for the Y values go through our point 5.5, negative nine and then continue to negative infinity for the Y values along our vertical asom tode X equals six. So to the right of our vertical asom tote X equals six, we can draw this, it's heading down toward negative infinity for the Y values. We can draw it coming up through our point 6.5, negative nine. And then it's going to continue along that vertical Asymptote X equals six. Then it's going to go through our X intercept at 70, then through our point of 8 9/4 and then it's turning it's heading toward positive infinity for the X values along the horizontal Asymptote Y equals three. And there's our graph well done. We'll catch you on the next one.

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Key Concepts

Horizontal Asymptotes

Vertical Asymptotes

X-Intercepts

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