Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-...

Question

Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)

Accepted Answer

Hello. In this video, we are going to be drawing the graph for the given function. The function that is given to us is G of X is equal to two X minus four divided by X plus five. So how do we start graphing a rational function? Well, in order to graph a rational function, there are a few things we need to find. We will need to find the vertical and horizontal asymptotes. The X and Y intercepts and determine the behaviors of the graph. So what we are going to do is we are going to start by finding the vertical asymptotes. Now how do we start finding the vertical asymptotes? Well, the vertical asymptotes represent the restrictions of the domain of a rational function. In this case here, what we want to do is we want to take the quantity in our denominator which is X plus five and find the values that make it zero. So we set up this equation which is X plus five equal to zero. And now we are going to solve for X in order to find our vertical Asymptote. In order to solve for X, we are going to subtract five to both sides of the equation. And that is going to give us X is equal to negative five. This is going to be the equation of our vertical Asymptote. Now, what we want to do is you want to find the horizontal Asymptote. In order to find the horizontal Asymptote, we are going to take a look at the leading exponents of the numerator and denominator. Now in our equation, the leading exponent of the numerator is one and the leading exponent of the denominator is one as well. Since the leading exponents are the same of both the numerator and denominator, the horizontal Asymptote is going to be equal to the ratio of the leading coefficients of both the numerator and denominator. So the leading coefficient of the numerator is going to be two and the leading coefficient of the denominator is going to be one that will give us a horizontal Asymptote of two divided by one which will simplify to Y is equal to two. So this is going to be the equation to our horizontal Asymptote. Now, what we want to do is we want to find the X intercepts. Now the X intercepts occur when the equation is equal to zero. So one shortcut that we have is that we know that the denominator cannot be zero because that will be undefined value. But if the numerator is zero, that will cause the entire function to become zero. So in order to find the X intercepts. We are going to take our numerator which is two X minus four, set it equal to zero and solve for X solving for X, we will add four to both sides of this equation that will leave us with two, X is equal to four. And if we divide both sides of the equation by two, that will leave us with X is equal to two. What this means is that we have an X intercept located at two comma zero. Next, what we want to do is we want to solve for our Y intercepts. In order to solve for the Y intercept, we want to find the value where Y is equal to zero. So we are going to input zero into our function and then solve. So G of zero will give us the function two multiplied by zero minus four divided by zero plus five. This will simplify to leave us with negative four, divided by five. And what this means is that we have a Y intercept at zero comma negative four divided by five. Finally, we want to determine the behaviors of the graph. In order to determine the behaviors of the graph, we are first going to create a number line. What we are going to place on this number line is our X intercepts and the vertical asymptotes. Now earlier we solved that the X intercept is located at two comma zero and we have a vertical Asymptote at negative five. So we are going to plot those two points on our number line. Next, what we want to do is we want to determine where the graph is increasing and decreasing. So what we could do is we can pick testing points between the regions of negative five and two, plug that into the value into the function and determine what type of value we get. For example, we can take the testing point to the left of negative five, which will be negative six. If we plug this to the function, we get two multiplied by negative six minus four, divided by negative six plus five. This will simplify to give us the value of 16. But notice that the value of 16 is positive. That tells us that the behavior of the graph to the left of negative five is going to be increasing. We will repeat this process by picking another testing point between negative five and two. A testing point we can pick is zero. But we already know that if we plug in zero to the function, we get the negative number negative four divided by five. And that tells us that the graph is going to decrease between negative five and two. Finally, we are going to pick a testing point to the right of two. And we can use the value of three. If we plug in three to the function, we get two multiplied by three minus four. Is divided by three plus five. And this will simplify to give us the value positive two, divided by eight. Since this is a positive number, this tells us that the graph will be increasing to the right of positive two. Now, all we need to do is we need to pick one of the options that matches all the information we just solved for. And again to recap, we have a vertical Asymptote located at negative five. We have a horizontal Asymptote located at positive two on the Y axis. We have an X intercept at two comma zero, we have a Y intercept of negative four comma five. And we understand that the graph is going to be increasing on both the left and right hand side and decreasing between negative five and two. The option that best fits these characteristics is going to be option B we have the X intercept at 20 Y intercept at negative 45. We know the graph is going to be increasing to the left of the X of the vertical Asymptote. And we know that it is increasing and decreasing around positive two. With that being said B is going to be the solution to this problem. So I hope this video helps you in understanding how to graph irrational function. And I'll go ahead and see you all in the next video.

Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)

Key Concepts

Rational Functions

Asymptotes

Intercepts

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