r/HomeworkHelp • u/random-burntout-user Secondary School Student • 1d ago
High School Math [Grade 10 Math] Explain this answer
Where did the -24+22 come from at the very beginning?
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u/Alkalannar 1d ago
A lot of times, things can become clearer if you simplify a rational function as much as possible.
For instance, rewriting (6n - 2)/(n - 4) as 6 + 22/(n - 4).
How do you do that? Polynomial long division, or adding appropriate forms of 0.
6n/n = 6, so 6 is the first term of our quotient.
(6n - 2) - 6(n - 4) = 6n - 2 - 6n + 24 = 22
22/n doesn't factor, so 22 is our remainder: 6 + 22/(n - 4)
Alternately, you could do (6n - 2 - 22 + 22)/(n - 4). Why 22? Because you want to have a multiple of n-4, namely 6n-24. And -22 is what you need to do to get that. But then you need to add 22 so you don't change the value.
Then (6n - 24)/(n - 4) + 22/(n - 4) = 6 + 22/(n - 4). Just as before.
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u/random-burntout-user Secondary School Student 21h ago
That's very helpful! My teacher barely broke down the process so this is much easier
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u/Alkalannar 9h ago
Glad I could help!
I actually find this very useful to separate out rational functions as much as possible. You can easily see transformations from a parent function (which you're at at the moment). In calculus, it can be much easier to take derivatives or integrate.
Example of transformations:
If your parent function is f(x), then your transformed function is af(b(x-c)) + d where:
a is the vertical stretch
b is horizontal compression
c is the shift to the right
d is the stretch up.Here, we have 22/(n-4) + 6.
The parent function is 1/n.
a = 22 is the vertical stretch.
b = 1 means there is no horizontal compression.
c = 4 means a shift to the right of 4.
d = 6 means a shift up of 6.And that is by no means evident in the form (6n - 2)/(n - 4).
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u/Southlander24 👋 a fellow Redditor 1d ago edited 1d ago
You're looking to break 6n - 2 apart into a multiple of n - 4 and a remainder term. Since there's a 6n, then 6(n - 4) = 6n - 24. And then to not change the 6n - 2 on top, you have to add back 22.
The form f(n) = a + b/(n - c) is very useful in general, where a, b, c are numbers. For instance, a = 6 is the horizontal asymptote of this rational function. That means for all n < 4, f(n) < 6 and that for all n > 4, f(n) > 6. There's other uses of this form in calculus where this form is easier to integrate than the original.