Use a graphing utility to graph the curve and the tangent line...

Question

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

Accepted Answer

Welcome back, everyone. Using a graph in utility, graph to function Y equals 3 X + 4 divided by 2X minus 5, and the tangent line to the curve at X equals -1. Now, to use our graphing utility to graph the function, we already have the equation of the function that we can simply enter into our utility. But for us to graph the tangent line to the curve at X equals -1, we'll need its equation as well. So how can we find the equation of the tangent line to the curve? Well, recall. That for any line we can write its equation in point slope form that says Y minus Y1 equals M multiplied by X minus X1. But to do that, we'll need two points on our, we'll need a point on our graph, X1 Y1, and we'll need the slope M. So if we can find the slope and the points for our graph of Y, then we'll be able to use it to find the tangent line to the curve at X equals -1, and then we can put that equation into our graphing utility to plot our graph. So let's go ahead and do that. First, let's start by trying to figure out the slope. What do we know that can help us? Well, we know that the slope of the line tangent to the curve at XFF X is the derivative of the function F of X, the derivative of FF X. So if we can differentiate Y, OK, and then substitute the value of X to be negative 1, then we'll get the slope. So let's go ahead and do that. Let's try that. So now we know that Y equals 3 X + 4 divided by 2X minus 5. So how will we differentiate Y? We'll notice here that Y is a quotient, OK? And we can differentiate quotients using the quotient rule of differentiation. The quotient rule basically says that if we have a function Y equal to U divided by V, where U and V are functions of X, then the derivative of Y with respect to X is going to be equal. So VU minus UV all divided by V2. In other words, if we let our our numerator be U, OK, 3 x + 4, or denominator be V X minus 5, then we can differentiate both of those and substitute them into our formula for the quotient rule to find the derivative of Y, OK? So, let's let Y be equal to, sorry, let you. B equal to 3 x + 4. OK. And we'll be. I don't know why I keep writing Y V to be equal to 2 X minus 5. If that's the case, then the derivative of you would be 3, OK. The derivative of 3 X is 3 and the derivative of 4 is 0, and the derivative of V is going to be 2, using the same idea here. In that case, know that we have our derivatives, if we substitute that into our function. Then the derivative of Y, OK, with respect to X is going to be equal to. V2X minus 5 multiplied by U 3. Minus, sorry, U, which is 3 X + 4 multiplied by V which is 2. I know all of this is being divided by 2X minus 5 squared. That's the case, if we expand here, we're going to get 6 X minus 15, OK, minus 6 X minus 8. In our numerator And when we simplify that some more, we're going to end up with negative 23 divided by 2 X minus 5R 2. Know that we have the derivative of Y, we can evaluate the derivative at X equals -1. There no at X equals -1. Then the derivative which is going to be our slope. Is going to be equal to -23. Divided by 2 multiplied by -1 minus 5, OK. All squared. Now, if we work at all, that's gonna be 23 divided by -7 squad and that equals -23 divided by 49. This is the slope of our graph or the slope of our line tangent to the curve, sorry, at X equals -1. Now what is the point? How are we gonna get X1 and Y1? Well, we can also tell that at X equals -1, we could get the Y value by simply substituting -1 into our function. So that's gonna be 3 multiplied by -1 + 4. I mean they both share the same point divided by 2 multiplied by -1 minus 5. OK. And now when we do that, 3 multiplied by -1 is -3, 3 + 4 is 1, and we're gonna have -2 minus 5, which equals -7. So at X equals -1, Y would be equal to -17. Now that we have MX1 Y1, we can substitute it into our point slope form, OK. Which now means that this is gonna be Y minus Y1 -17. Eals M23 49th multiplied by x minus X1, so it's minus -1. If we simplify that some more here, I think I might have to uh make a little space. Then we're gonna get Y + 1/7 to be equal to negative 2349 of X, OK. Minus 23 49th. OK. And no if we subtract 1/7 from both sides, then Y is gonna be negative 2349 of X, OK, minus 2349 minus 17, which gives us negative 3049. OK. Thus, this is the equation of our tangent line. Our tangent line is defined by Y equals -2349 X minus 3049. No, we can input our equation Y equals 2 x + 4 divided by 2X minus 5 and the equation for the tangent line in our graphing utility. Now when you do that, I'm just gonna paste the graph here, but you should get a graph that looks like this, OK? Where the line in blue is our tangent line and the lines in black are our original graph of Y. Therefore, this is what our graph would look like by using a graphing utility. Thanks a lot for watching everyone. I hope this video helped.

Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = (x + 5) / (x - 1); a = 3

Key Concepts

Graphing Rational Functions

Tangent Lines

Using Graphing Utilities

Watch next