Table 1 gives several positive –values that are successively halved

MATHEMATICS · Multi-part · AoPS · Mathematics · Proportion · Proportion

Problem

Table 1 gives several positive

x

–values that are successively halved. For each

x

, the corresponding

y

for the function

y=\dfrac{1}{x}

is to be found. (a) Table 1 |

x

4

2

1

\tfrac12

\tfrac14

\tfrac18

| |-----|-----|-----|-----|---------|---------|-----------| |

y

| ? | ? | ? | ? | ? | ? | Complete the table by calculating every missing

y

–value as a fraction. (b) (ii) For each ordered pair

(x,y)

obtained, compute the product

xy

. Record these products in a separate row beneath the table. (c) (iii) What pattern do you notice in the products

xy

? State your observation clearly. (d) (iv) Using the completed table, describe in words how the value of

y

changes when

x

is repeatedly halved. Relate this behaviour to the overall shape of the graph of

y=\dfrac{1}{x}

in the first quadrant.