Graph each rational function. See Examples 5–9. ƒ(x)=(6-3x)/(4-x)

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Hello. Today we're gonna be graphing the following rational function. So what we are given is F of X is equal to eight minus two X over six minus X. So the first thing we want to do when graphing the rational function is to first identify any restrictions on the function itself. Or in other words, we're going to be looking for the vertical ascent tos of X C four, a rational function. The only value we're not allowed to have in the denominator is zero because if the denominator is zero, then the whole function is going to be undefined. So in order to get the vertical asymptotes, we're going to be taking our denominator which is six minus X and setting it equal to zero. And if we solve for X, that is going to be the graph of our vertical ascent tote. Now, in order to solve for X, we can add X to both sides of this equation and that is going to give us X is equal to positive six. So that is going to be the vertical ascent to for a rational function. Now, what we wanna do is we want to go ahead and identify our horizontal ascent to. Now, let's take a look at the leading X terms. We're gonna quickly rewrite the given rational function by putting the X terms first, that is going to give us negative two X plus eight over negative X plus six. If we take a look at the exponents of the leading X terms, the values of the exponents are exactly the same in both the numerator and denominator. And that means that the horizontal ascent tode is going to equal to the rational quantity of the leading coefficients of the X terms. Now, the leading coefficients of the X terms is going to be negative two over negative one. And if we simplify this, this is going to give us 2/1, which just gives us two. So that means that we have a horizontal Asymptote at Y is equal to two. Now, what we want to check is to see if the graph is going to intercept the horizontal sy to. So in order to check that we're going to need to set our function equal to the value of the horizontal asym tode that is going to be eight minus two X over six minus X is equal to two. In order to solve for X, we're gonna multiply both sides by the denominator that is going to give us eight minus two, X is equal to two times the quantity six minus X. We're gonna go ahead and distribute two into the quantity six minus X. And that is going to give us eight minus two, X is equal to 12 minus two X. We can go ahead and add two X to both sides of the equation to cancel out the X terms. And what we are left with is eight is equal to 12. Now this is an inconsistent equation. And since we got an inconsistent equation, that means that the horizontal asy tote is not going to be intercepted. So now that we know that the horizontal as to is not intercepted, the next thing we wanna do is solve for the X intercepts. In order to solve for the X intercepts, we're going to set the equation equal to zero and solve for X. So that's gonna be eight minus two, X over six minus X is equal to zero. If we multiply both sides by the denominator, what we are left with is eight minus two, X is equal to zero, we can subtract eight to both sides to give us negative two, X is equal to negative eight. And if we divide both sides by negative two, what we are left with is X is equal to four. What this means is that we have one X intercept and X is equal to four. And finally, we need to get the Y intercept in order to get the Y intercept, we're just going to set the values of X equal to zero. That is going to give us Y is equal to eight minus zero over six minus zero. And this is going to simplify to give us 4/3, 4/3 is approximately equal to about 1.3. So that means that we have a Y intercept at zero comma 1. approximately. And finally, we need to get the regions of increasing and decreasing. In order to do that, we're going to take a number line. This number line is gonna go from negative infinity to positive infinity. And what we're gonna put on this number line is going to be our vertical asymptotes and our X intercepts, we have one X intercept and one vertical ascent tode. So we're gonna be pointing four and six on the number line. Now, what we wanna do is you want to take a testing point from each of these regions to determine if the graph is going to be increasing or decreasing. So let's go ahead and take a testing point going from negative infinity to four. The easiest testing point to take here is going to be zero. So if we plug in zero into our function, we get the alpa value of 4/3. Now, since this is a positive value, that means that this graph is going to be increasing from negative infinity to four. Now, we need a testing point between four and six and that testing point can be five. So if we plug in five into our function, the output value is going to be negative two. Since this value is negative, the graph is going to be decreasing in this region. And now we need a testing point going from six to infinity, we can go ahead and choose seven. And if we plug in seven into the function, we get the value of six. And since this is a positive value, the region is going to be increasing. So now we have all the necessary information we need in order to graph the rational function, we have the vertical ascent totes, we have the horizontal ascent totes, we determined that the horizontal aceto is not going to be intercepted. We found the X intercepts, we found the Y intercept and we found the regions of increasing and decreasing. So now we just need to put this together on an, on the actual graph. So we're going to be drawing an axis. Now keep in mind let's go ahead and first label the vertical Asymptote and horizontal Asymptote. So we know that the vertical Asymptote is going to be at X is equal to six. So we can go ahead and represent that with a dotted line and we know that the horizontal ascent tode is going to be Y is equal to two. So we can say that Y is equal to two is roughly about here. Next, we have the Y intercept at zero comma 1.3. So that's gonna be roughly about here. And we have an X intercept at the value a positive four. So we have X intercept at four comma zero. Now keep in mind we can kind of use these intercepts as a guiding, as guiding points for the shape of the graph. See we know that the graph is going to be increasing, going from four to positive infinity, which means that the graph is going to have to increase to the Y intercept and keep increasing to the left. However, this graph is never going to intercept the horizontal ascent to it's always gonna stay underneath. Then from up above, we know that the graph is going to decrease as it approaches six. So as the graph approaches six, the graph is going to decrease but not touch the vertical asy tode. And we know that the graph is going to be increasing going from six to positive infinity. And that means that the next region of the graph is going to lie above the horizontal ascent to and is going to have a rough shape like this. And this is going to be the rough shape of the graph of the rational function. Now keep in mind you don't have to be super accurate with this because calculus is required to find any trema for any type of values. And with that being said, with that being said, I hope you understand 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)=(6-3x)/(4-x)

Key Concepts

Rational Functions

Asymptotes

Intercepts

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