Inverse Variation.

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What You Will Learn Recognize and solve direct and joint variation problems Recognize and solve inverse variation problems.

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Presentation transcript:

Inverse Variation

Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people. No. of people (x) 1 2 3 4 5 … Amount (y mL) 1500 750 500 375 300 How does y change when x increases?

Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people. No. of people (x) 1 2 3 4 5 … Amount (y mL) 1500 750 500 375 300 When x increases, y decreases.

Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people. No. of people (x) 1 2 3 4 5 … Amount (y mL) 1500 750 500 375 300 Do you have any further observations about the relation between x and y?

It is observed that the value of xy is a constant. No. of people (x) 1 2 3 4 5 … Amount (y mL) 1500 750 500 375 300 xy 1500 1500 1500 1500 1500 … It is observed that the value of xy is a constant. That is, xy = 1500 (or ). y = x 1500 We say that the amount of soft drink (y mL) obtained by each person varies inversely as the number of people (x). Such a relation is called an inverse variation (or inverse proportion).

Inverse Variation x k y = then xy = k or , In general, if y varies inversely as x,  ‘y varies inversely as x’ can also be written as ‘y is inversely proportional to x’. x k y = then xy = k or , where k is a non-zero constant. k is called the variation constant. In symbols, we write . x y 1 µ

Consider any two quantities x and y which are in inverse variation. i.e. , where ¹ k x y = The graph of y against x is a curve like the one shown below. As x tends to 0, the curve gets closer to the y-axis. y x O k = The curve does not intersect the axes. As x increases, the curve gets closer to the x-axis.

Follow-up question If d varies inversely as n, and d = when n = 4, find (a) the equation connecting d and n, (b) the value of n when d = 8. 2 1 (a) ∵ d varies inversely as n. ∴ , where k ¹ 0 n k d = k is the variation constant. By substituting n = 4 and d = into the equation, we have 2 1 4 2 1 = k 2 = k

Follow-up question If d varies inversely as n, and d = when n = 4, find (a) the equation connecting d and n, (b) the value of n when d = 8. 2 1 ∴ 2 n d = (b) When d = 8, 2 8 = n 4 1 = n