In Exercises 57–80, follow the seven steps to graph each rational...

Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x^2+x−12)/(x^2−4)

Accepted Answer

Graph. The given rational function following the seven steps, F of X equals X squared plus X minus 30 divided by X squared minus 36. Now, to solve this, I'm first going to simplify our functions. We have FX X squared plus X minus 30 divided by X squared minus 36. We can factor both of these polynomials was on top, we get X plus six multiplied by X minus five. On the bottom, we have X plus six multiplied by X minus six based on difference of two squares that simplifies to X minus five divided by X minus six. We can now use our seven steps. Our first step is determine our symmetry, determine our symmetry. We will take F of negative X and see what we get out. FFF negative X gives us the original equation back out. We have Y axis symmetry if it gives us the original equation but negative, we have origin symmetry. We can go ahead and use our simplified function and instead plug in negative X for X. So we have negative X minus five divided by negative X minus six. This is neither form we want. So this actually has no symmetry. Our second step will be to determine the Y intercept. For this, we will let X equals zero FF zero will then be zero minus five divided by zero minus six, which is negative five divided by negative six, which gives us 5/6. So our Y inter set is 056. Our third step will be to find our X intercept. Our X intercept happens with Y equals to zero. So it will say zero equals X minus five divided by X minus six, multiply both sides by X minus 6 to 0 equals X minus five. Solving for X, we get X equals positive five, meaning our X intercept happens at 50. For step four, we need to find our asymptotes. In particular. We want to find our vertical Asymptote. For the vertical Asymptote, we will set our denominator equal to zero and then solve for X, our denominator X minus six and we set it equals to zero to get X equals six. When solving for X for step five, we want our horizontal as to the horizontal as is based on the degree of our rational function. Now we notice for horizontal acid, the degree of our numerator is the same as the degree of our denominator because they had the same degrees a horizon Lato then will be the ratio of the coefficients. So we have X minus five divided by X minus six. That means our coefficient for each of these numbers is one. So we have one divided by one, which is just one. In other words, our horizontal Aspin tode happens at Y equals one. Finally, for step six, let's find values on the graph. He'll make a table XF of X. Let's say we take negative four, negative two zero 58 and 10. If we were to plug all these in, we would get, let's say 0.9 0. 0.833 01. and 1.25. Finally, our seventh step is to use all of our information to graph the function. If we look at all of our possible graphs, we would then determine the answer to our problem. Most closely follows answer B B has all the information that we found. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x^2+x−12)/(x^2−4)

Key Concepts

Rational Functions

Asymptotes

Graphing Steps

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