Sketch the graphs of the rational functions in Exercises 53–60...

Question

Sketch the graphs of the rational functions in Exercises 53–60.

y= (x + 1) / (x - 3)

Accepted Answer

Sketch the graph of the function. The 1st and 2nd derivative are given, where we have F of X equals X minus 2 divided by X + 4, and we also have F of X being 6 divided by X + 4 squad, and F of X being negative 12 divided by X + 43. We're also given a graph to graph or function. Now the graph this, we need to find some information. Let's first find the domain. The domain, we noticed we have X minus 2 divided by X + 4. And so, to check our domain, we will check our denominator. Our denominator cannot be equals to zero. So, if we were to solve this, we see that by subtracting 4, We get X cannot be equals to. -4 That tells us our domain. Will be negative infinity to -4 and -4 to infinity. Now let's check the symmetry. To check symmetry, we are going to replace our X with negative X and see what happens. So, We have negative X minus 2, divided by negative X plus 4. Which gives us X minus 2. Divided by negative X plus 4. Now, this doesn't follow our symmetry rules, so this is neither. Even or on. Now, let's check it's increasing and decreasing intervals. ever increasing and decreasing intervals. To check this, we'll look at our first derivative. We have 6 divided by X plus 4. Squared Now, X + 4 squared is always greater than 0, except for When X equals -4. This means that this function is always increasing except for where X equals -4. So, we can say negative infinity to -4. And -4 to positive infinity are increasing. Next, we can check the concavity. We look at our 2nd derivative to determine this, so we have our concavity. We have 12, divided by X + 4 raised to the 3rd hour. Now, the sign of This 2nd derivative is dependent on the sign of our denominator, X + 43rd power. So, we have X + 4 race to the 3rd. This is positive. When X is greater than. Neither 4 And this is negative. When X is less than -4. That means that for the first part of this, greater than -4. This is concave down. And the other part is concave up. This tells us there's a change in concavity at -4, but we know that this is the asymptote, so it's actually not gonna matter in this case. Next, let's find the intercepts. To find the innocent. We will set our equation equals to 0 to find the X intercept first. We have 0 equals. X minus 2, divided by X plus 4. We can multiply on both sides to just get 0 equals X minus 2. And if we add 2, You get X equals 2. Meaning we have an intercept. At the 0.20. We can also check the Y intercepts. By letting X equals 0. We have 0 minus 2 divided by 0 + 4. Which gives us -2 divided by 4 or -12, meaning our Y intercept. Is 0-12. No. We can finally check our asymptotes and end behavior. Have our asymptote Which we would look at the denominator of this function to find the vertical one. So for vertical We have X + 4 equals 0, or X equals -4. And for the horizontal, we look at Our rational function as a whole. Because the degrees of our polynomials are the same on the numerator and the denominator, Our asymptote occurs at Y equals 1 divided by 1, or the coefficient ratio of our function. Which is just what? And finally, we can check the end behavior. By checking the limit. The limit as X approaches plus or minus infinity of our function. And this is actually the same value, which is just one. So now we have everything we need. To graph this function. Now, Remember, we have our white intercepts. At 0, 1/2 in our X intercept at 20. We have a vertical asymptote. At X equals -4. And the horizontal asymptote. At Y equals 1. Finally, we need another point. And so, for example, we could plug in. -6, which would give us a wide output of 4. Now we can make a graph Let us draw a smooth curve between our points following the asymptote. And that is our graph to this problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Sketch the graphs of the rational functions in Exercises 53–60.
y= (x + 1) / (x - 3)

Key Concepts

Rational Functions

Vertical Asymptotes

Horizontal Asymptotes

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