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Mth 163
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** Question Form_labelMessages **
Assignment 4:Function notation
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when evaluating y(vertex) = f(x vertex)
= b^2 / (4a) - b^2 / (2a) + c
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to subtract the fractions we must use the least common denominator of the two figures which in this case is 2. Do we multiply both fractions by 2/2 or just 1 of them, if so which one?
In the notes you multiplied only the second fraction by 2/2 giving:
b^2 / (4a) - 2 b^2 / (4a) + c
From here b^2 is subtracted from -2b^2 giving -b^2 for the numerator. For the denominator, in the notes you have (4a) as the answer, but if we subtracted 4a from 4a would we not get 0?
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I must be missing something here amidst the algebra.
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When you add or subtract fractions they have to have the same denominator; you then add or subtract the numerators, but you don't add or subtract the denominators.
A simple example will show that it is so:
If you cut a pie into 8 pieces and someone gets 5 pieces while the other gets 3, then the first person has 5/8 of the pie while the second has 3/8. The difference between these numbers is 2/8 (which of course reduces to 1/4).
So 5/8 - 3/8 = (5 - 3) / 8 = 2/8.
Notice that while we subtracted the numerators, we didn't subtract the denominators.
When adding ot subtracting fractions we first express in terms of a common denominator, then subtract the numerators.
We can understand on a deeper level why this must be so, by thinking in terms of the distributive law:
Formally we could do our calculation as follows:
5/8 - 3/8 = 1/8 * ( 5 - 3 ), by the distributive law.
Thus
5/8 - 3/8 = 1/8 * (5 - 3) = 1/8 * 2 = 2 / 8.
Similarly b^2 / (4 a) - 2 b^2 / (4 a) could be written using the distributive law as
1 / (4 a) * (b^2 - 2 b^2) = 1 / (4 a) * ( -b^2) = -b^2 / (4 a).
Another way to simplify the same expression, also using the distributive law:
b^2 / (4 a) - 2 b^2 / (4 a) = (1 - 2) * b^2 / (4 a) = (-1) * b^2 / (4 a) = - b^2 / (4 a).
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