Question

Draw a graph for the following table and identify the variation.

\[x\]	2	3	5	8	10
\[y\]	8	12	20	32	40

 Hence, find the value of \[y\] when \[x=4\].

Answer 1

Hint:we will use the variation table to first draw the graph helping us to determine the equation of the line. from the equation we will find the value of which we need to find later on.


Complete step by step solution:
From the given table we can deduce that as \[x\] increases the value of \[y\] also increases. Also notice that the value of \[y\] is increasing in a systematic manner. Thus, we will check by the most simple variation….
Let \[y=\text{K}x\].
\[\dfrac{y}{x}=\text{K}\]
Where K is the constant of probability.
Thus, from the given values ,we can deduce that
\[\text{K=}\dfrac{y}{x}=\dfrac{8}{2}=\dfrac{12}{3}=\dfrac{20}{5}=\dfrac{32}{8}=\dfrac{40}{10}=\dfrac{4}{1}\]
Thus, we get the relation \[y=4x\] easily which is similar to an equation of a straight line. So we will compare this equation with the general equation of straight line which is
\[y=mx+c\,\,\,\,\,\text{ }\!\![\!\!\text{ where }m\,\,\text{is slope and }c\,\text{is }y-\operatorname{int}ercept\text{ }\!\!]\!\!\text{ }\].
And we get the values….
\[m=4\,\,\,\,\text{and }c=0\]

From this we know that slope is 4. And since the slope is positive it will constantly move upward.
 We can say that the line does not intersect with \[y\]-axis since it does not have a \[y\]-intercept.
And thus the plotting can now be done with the points (2,8)(3,12)(5,20)(8,32)(10,40) we got from the variation table.
Moreover, by putting the value \[x=4\] in the equation \[y=4x\] We get the value of \[y\] as 16. Therefore , the point is (4,16).
The line will pass from the origin(0,0).

Note: if the equation would have been more complex , such as, having a constant term as well \[y=kx+L\text{ }\!\![\!\!\text{ L is a constant variable }\!\!]\!\!\text{ }\], then we would have also be given more factors from the question to determine what the value of this constant term could have been. Also make sure to plot the points given from the variation carefully.