Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+8x+16)/...

Question

Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+8x+16)/(x^2+4x-5)

Accepted Answer

Hello, today we're going to be graphing the following rational function. So the rational function that is given to us is F of X is equal to X plus eight times X minus five over X plus times X minus two. Now, the first thing we're gonna want to do in order to graph this rational function is to first find the vertical asymptotes. Now, currently, we are working with the rational function and the only value that causes the rational function to be undefined is going to be zero. So we don't want any factors of X in the denominator to equal to zero. Otherwise we get an undefined value. So for the vertical as totes, we're going to take the quantities X plus 11 times X minus two and set it equal to zero using the zero property, we can go ahead and set both of the factors of X equal to zero. That is going to give us explicit 11 is equal to zero and X minus two is equal to zero. And solving for X in both of the equations is going to give us X is equal to negative 11 and X is equal to two. So we have two vertical ascent totes. For the rational function, we have X is equal to 11 and X is equal to two. Now, we need to determine what the horizontal ascent tode is going to be. Now, in the numerator, if we were to multiply the two quantities together, we get a polynomial whose leading coefficient is X squared. And in the denominator, if we multiply the two quantities together, we get a polynomial whose leading coefficient is X squared. Since the exponent of the numerator and the denominator have the same value, that means that the horizontal ascent tote is going to equal to the rational quantity of the leading coefficients. Now, in this case here, the leading coefficients are gonna be one in one, which means that the horizontal ascent tode is going to be represented as Y is equal to one. Now, we need to determine the X intercepts for the rational function. Now, the X intercepts occur when the rational function is equal to zero. And that means that if the numerator becomes zero, then the entire function becomes zero. So what we're going to do for the X intercepts is take the quantities in the numerator which is X plus eight times X minus five and set it equal to zero. And just like the vertical ascent totes, we can set both of the quantities of X equal to zero that is going to give us X plus eight is equal to zero and X minus five is equal to zero. And solving for X, we get X is equal to negative eight and we get X is equal to positive five. So these are gonna be the X intercepts for a polynomial. I mean for the rational function. Now we need to check to see if the function is going to intersect the horizontal Asy Tode. So we know that the value of the horizontal Asy Tode is going to be Y is equal to one. And if it intersects the horizontal ascent tode, then the function has to equal to one. So we're going to have X plus eight times X minus five over X plus 11 times X minus two equal to one. So now we need to sell for X, we can start by multiplying both sides by the denominator. And if we multiply both sides by the denominator, we get X plus eight times X minus five is equal to X plus 11 times X minus two. Now, what we wanna do is we want to fold together both sides of the equation that is going to give us X squared plus three, X minus 40 is equal to X squared plus nine X minus 22. In order to solve for X, we can subtract X squared to both sides of the equation and we can move X to one side of the equation. So we can subtract nine X to both sides and we're gonna move all the constants to one side of the equation. So we're gonna add 40 to both sides of the equation. So our XX square terms are going to cancel out. And by moving the terms around what we are left with is negative six, X is equal to 18. And finally, if we divide both sides by negative six, we get X is equal to negative three. So what that means is that the graph is going to intercept the horizontal ascent tode at the point negative three comma one. And now finally, we want to go ahead and determine the regions of increasing and decreasing for the rational expression for the rational function. So we're going to draw a number line and this number line is gonna go from negative infinity to positive infinity. Now what's gonna go on this number line is going to be the vertical asymptotes and the X intercepts. So that's going to be negative 11, negative eight, positive two and positive five. And now we want to take a testing point from each of the regions to plug into the function and determine whether the regions are going to increase or decrease. If we were to take a testing point from the first region going from negative infinity to negative 11, we can choose negative 12 as the testing point. And if we plug in negative 12 into our function, we get 34/7, which is a positive value that means that the function is going to increase in this region. Now we want a testing point between negative 11 and negative eight, we can go ahead and choose negative 10. And if we plug in negative 10 into our function, we get the output negative 5/2. Since this is a negative value, that means that this region of the graph is going to decrease, then we want a testing point between negative eight and two. We can go ahead and choose zero and plugging in zero into our function is going to give us the output of 20/11. This is a positive value. And that means that the region is going to increase between negative eight and two. Furthermore, because this is a zero value for X 20/11 is also going to be our Y intercept. So the Y intercept is going to be 20/11. Next, we want to choose a testing point between two and five and we can go ahead and choose the value of three. So if we plug in three into our function, we get negative 11/7, this is a negative value. So the region of this graph is going to decrease. And finally, we need a testing point from five to positive infinity and we can go ahead and choose six. So if we were to plug in six into our function, we get the output value of 7/34. And since this is a positive value that means that this region is going to be increasing. So we got all the necessary pieces of information to graph the rational function. We have our vertical asymptotes, we have our horizontal asymptotes, we have our X intercepts, we know where the graph is going to intercept the horizontal Asymptote. We have the Y intercept and we have the regions of increasing and decreasing for the graph. Now, we just need to go ahead and select the graph that accurately represents all this information. And that graph is going to be option A just to verify we have the vertical ascent totes at X is equal to negative 11 and X is equal to positive two. We have the X intercepts at negative eight and positive five and we have a Y intercept at 20/11 and the regions of increasing and decreasing match the regions that we came up with. And we have a horizontal aceto at Y is equal to one. So with that being said, the answer to this problem is going to be a. 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)=(x^2+8x+16)/(x^2+4x-5)

Key Concepts

Rational Functions

Graphing Techniques

Asymptotes

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