Graph each rational function. See Examples 5–9. ƒ(x)=5x/(x^2-1)

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Hello. Today we're going to be graphing the following rational function. So the rational function given to us is F of X is equal to 13 X over X squared minus 25. Now, the first step into graphing this equation is to go ahead and first get the vertical asymptotes. Now, the vertical ascent totes represent any restrictions that we have on the graph. And the for the rational function, the only restriction we have is that we cannot allow the denominator to be zero. If the denominator is zero, then that means the function is undefined. So what we're going to do is we're going to take X squared minus 25 which is our denominator and set it equal to zero in order to find our vertical ascent totes. In order to solve for X, we're first going to add 25 to both sides of the equation that is going to give us X squared is equal to 25. And if we take the square root of both sides of the equation, we get X is equal to plus or minus five, that means we have two vertical asymptotes. For the rational function, we have X is equal to positive five and X is equal to negative five. So those are gonna be the two restrictions that exist for our rational function. Now we want to solve for the horizontal Asymptote. Now one thing to note about the leading coefficients is that we have X over X squared, the exponent in the denominator is greater than the exponent in the numerator. And whenever you have a case where you have a rational function whose exponential values are greater in the denominator than the numerator, then you can automatically assign the horizontal Asymptote to be Y is equal to zero. So we know that the horizontal ascent to is going to be at zero. Next, we want to go ahead and determine if the graph intercepts the horizontal Asymptote. Now, for that, we know that the output of the graph needs to be zero. So for the interception, we're gonna have the function 13 X over X squared minus 25 equal to zero. If we multiply the denominator to both sides of the equation, what we are left with is 13 X is equal to zero and dividing both sides of this equation by 13 is going to give us X is equal to zero. So that means the graph is going to intercept the horizontal ascent tode at the 0.0 comma zero. Now, what we wanna do is go ahead and find any X intercepts in order to get the X intercepts we want to go ahead and solve for X in the graph. But notice that when we were checking for the interception of the horizontal ascent to, we set the graph equal to zero and we got an output of X for zero. That means that the X intercept for the graph is going to be X is equal to zero. Now, what we want to go ahead and do is find the regions of increasing and decreasing. In order to do that, we're going to draw a number line. This number line is going to go from negative infinity to positive infinity. And what we're gonna label the number line with is our vertical ascent totes and our X intercepts, there are only three values that we came up with two vertical asymptotes at negative five and positive five and X intercept at zero. Now, what we wanna do is we want to take a testing point from each of these regions to plug into our function and use that to determine whether the region is going to increase or decrease. So for the first region going from negative infinity to negative five, we can go ahead and pick a testing point of negative six. If we were to plug in negative six into our function, we get the output negative 78/11. Since this is a negative value that tells us that the graph is going to be decreasing in this region. Now we wanna pick a testing point between negative five and zero. And that can be negative one. So if we plug a negative one into our function, we get the output value of 13/24. Since this is a positive value, that means that this region is going to be increasing. Now, we want to pick a testing point between zero and five and that can be positive one. When we plug in positive one into our function, we get negative 13/24. Since this is negative, that means the region is going to be decreasing. And finally, we need a testing point going from five to positive infinity. And that can be six. So if we plug in six into our function, we get 78/11. And since this is positive, that means the region is going to be increasing. So just to summarize, we have all the information we need to graph the function. We have the vertical ascent totes at five and negative five. We have the horizontal ascent tode at Y is equal to zero. We know that the horizontal ascent to is going to be intercepted at zero, comma zero. We have the XX intercept and we have the regions of increasing and decreasing. Now we just need to use this information to select the correct graph from up above. And the graph that accurately matches all this information is going to be option D just to clarify. We have the X intercepts, I'm sorry, the vertical asymptotes at negative five and positive five, we have the horizontal ascent tote at Y is equal to zero and we have the Y intercept at zero comma zero. Furthermore, the regions of increasing and decreasing match the regions that we came up with. The graph is going to be decreasing as it approaches negative five. It's going to be increasing as it approaches negative five on the right. It's going to be decreasing as it approaches positive five on the left and increasing as it approaches positive five on the right. And with that being said, the answer to this problem is going to be D so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Graph each rational function. See Examples 5–9. ƒ(x)=5x/(x^2-1)

Key Concepts

Rational Functions

Asymptotes

Graphing Techniques

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