Question

A direct variation function contains the points (2, 14) and (4, 28). Which equation represents the function?

Answer 1

a direct variation function, is just another way to word a "linear equation", and the slope will then be the "constant of variation".

so, in short, what's the equation of the line that runs through 2,14 and 4,28?

$\bf \qquad \qquad \textit{direct proportional variation}\\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 2}}\quad ,&{{ 14}})\quad % (c,d) &({{ 4}}\quad ,&{{ 28}}) \end{array}$

$\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{28-14}{4-2}\implies \cfrac{14}{2}\implies 7 \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})}\implies y-14=7(x-2) \\\\\\ y-14=7x-14\implies y=7x$

Answer 2

we know that

A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form $\frac{y}{x}=k$ or $y=kx$

In this problem we have

Points $(2,14)$ and $(4,28)$

so

Find the value of k

First point

$\frac{14}{2}=k$

$k=7$

Second point

$\frac{28}{4}=k$

$k=7$

the function is

$y=kx$ --------> substitute the value of k

$y=7x$

therefore

the answer is

$y=7x$

In this problem with a single point was sufficient to calculate the equation, since in a direct variation the line passes through the origin, it is not necessary to use the formula of the slope