Name: Direct and Inverse Proportion
Uploaded: 2019-07-19
Duration: 5 min 10 s
Description: Quantities like price, speed, distance, etc., show confusing trends of increase or decrease to the other quantity. Let’s learn how to deal with these confusi...

Question

Determine which of these functions represent direct proportion, and find the constant of proportionality for those that do.

Accepted Answer

Given graph is

The given lines represent the function,

Since every point on the line satisfies the equation of the function. So, let us find the coordinates of the points that satisfies the equation of the lines.

For line $' a'$ : Few points that satisfies its equation are

$(- 1, - 3), (1, 3), (2, 6)$

We can observe that as the value of $x$ coordinate increases the value of $y$ coordinate also increases in the same ratio.

Since $\frac{- 1}{- 3} = \frac{1}{3} = \frac{2}{6} . . . . (i)$

Thus, the quantities, $x$ , and $y$ are in direct proportion.

$∴ x = k y$ , where $k$ is called the constant of proportionality, and the value of $k$ is $\frac{1}{3}$ [using $(i)$ ].

Similarly,

Few points satisfying line $b :$ $(- 2, - 4), (- 1, - 1), (1, 5)$

Since $\frac{- 2}{- 4} \neq \frac{- 1}{- 1} \neq \frac{1}{5}$

$∴ x$ , and $y$ are not in direct proportion.

Hence, the line $b$ does not represent the direct proportion.

Few points satisfying line $c : (- 4, - 2), (2, 1), (4, 2)$

Here we observe that as the value of $x$ -coordinate increases the value of $y$ -coordinate also increases in the same ratio.

Since

$\frac{- 4}{- 2} = \frac{2}{1} = \frac{4}{2} = 2$

Thus, the function $c$ represent the direct proportion, and the constant of proportionality is $2$ .

Few points satisfying line $d : (- 5, - 4), (5, 4)$

We observe that as the value of $x$ increases the value of $y$ also increases in the same ratio.

Thus, the function $d$ represent the direct proportion, and the constant of proportionality is $\frac{5}{4}$ .

Few points satisfying line $e : (- 5, - 4), (- 4, - 2), (- 2, 2)$

Here we see that

$\frac{- 5}{- 4} \neq \frac{- 4}{- 2} \neq \frac{- 2}{2}$

Hence, this function $e$ does not represent the direct proportion.

Few points satisfying line $f : (- 1, 4), (1, - 4)$

We observe that as the value of $x$ increases the value of $y$ also increases, and in the same ratio.

Thus, the function $e$ represent the direct proportion.

Hence the functions, $a, c, d,$ and $f$ represents the direct proportion, and the corresponding constants of proportionality are $\frac{1}{3}, 2, \frac{5}{4},$ and $- \frac{1}{4}$ respectively.

Determine which of these functions represent direct proportion, and find the constant of proportionality for those that do.

Important Questions on Getting More Done in Less Time (Direct and Indirect Proportion)

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