Solve each problem. Find a rational function ƒ having the grap...

Question

Solve each problem. Find a rational function ƒ having the graph shown.

Graph of a rational function with vertical asymptote at x=1 and horizontal asymptote at y=-3.

Accepted Answer

Hey, everyone here we are asked to find F of X by using the graph of the rational function F which is given below. Here we have four answer choice options. ABC and D, each of which have a different function for F of X. So to find F of X by using this graph, we just need to pull out the given information from the graph. So we can first notice that we are given an X intercept at the point for zero. And so with this being our only X intercept, we know that the numerator of our function must have the factor of X minus four. So we can begin forming our function by placing the quantity of X minus four in our numerator. Next, we see that we have a vertical Asymptote which is shown by this vertical dash line on our graph. And this is the line X is equal to five. Well, this vertical Asymptote tells us that the denominator of our function must have the factor X minus five. So in our function, we will just have the quantity of X minus five in the denominator so far. And now we are also given a horizontal Asymptote and this horizontal Asymptote is the line Y is equal to negative five. And from this horizontal Asymptote, we get that the leading coefficient of the numerator must be equal to negative five. So then for our function, we just need to include a negative five times some variable A since again, negative five is going to be the leading coefficient of the numerator. And so now the next thing we need to notice is that since we only have one factor in the numerator and that the horizontal Asymptote is Y is equal to negative five, then we know that the denominator will only contain one factor. So in the denominator, we will only have this quantity of X minus five. So here we have formed our function so far. So we have F of X is equal to negative five times A times the quantity of X minus four all divided by the quantity of X minus five. So now we need to find this value of A and we can do this by utilizing the Y intercept and the Y intercept is given as the 50.0 and negative four. And so from this point, we see that X is going to be equal to zero and that Y which is the same as F of X is going to be equal to negative four. So now we just need to substitute in these values for X and F of X into our function to find our value for A. So we have negative four is equal to negative five times a times the quantity of zero minus four all divided by the quantity of zero minus five. Now just simplifying we have negative four is equal to negative five times a times negative four divided by negative five. Continuing to simplify our product. In the numerator, we have negative four is equal to 20 times A all divided by negative five. Simplifying the 20 divided by negative five. On the right side, we get negative four is equal to negative four times A. Now getting rid of the coefficient of A, we divide both sides by negative four. And we see that our value for A is just positive one. So now substituting in this value for a into our function, we see that our final function is F of X is equal to negative five times the quantity of X minus four all divided by the quantity of X minus five. So with our final simplified function, we are left with answer choice A where again we have F of X is equal to negative five times the quantity of X minus four all divided by the quantity of X minus five. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each problem. Find a rational function ƒ having the graph shown.

Key Concepts

Rational Functions and Their Graphs

Vertical and Horizontal Asymptotes

Using Points on the Graph to Determine Function Parameters

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