To find the equation that represents the direct variation function containing the points [tex]\((2, 14)\)[/tex] and [tex]\((4, 28)\)[/tex], follow these steps:
1. Identify the general form of the direct variation equation:
[tex]\[
y = kx
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Find the constant of proportionality [tex]\( k \)[/tex] using the provided points:
Given the points [tex]\((2, 14)\)[/tex] and [tex]\((4, 28)\)[/tex], we know that:
[tex]\[
y_1 = 14 \quad \text{when} \quad x_1 = 2
\][/tex]
and
[tex]\[
y_2 = 28 \quad \text{when} \quad x_2 = 4
\][/tex]
3. Calculate the slope (constant of proportionality) [tex]\( k \)[/tex]:
The slope [tex]\( k \)[/tex] of the line passing through the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in direct variation can be found using the formula:
[tex]\[
k = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
4. Substitute the given points into the formula:
[tex]\[
k = \frac{28 - 14}{4 - 2}
\][/tex]
5. Perform the arithmetic operations:
[tex]\[
k = \frac{14}{2}
\][/tex]
[tex]\[
k = 7
\][/tex]
6. Form the direct variation equation with the calculated [tex]\( k \)[/tex]:
Substitute [tex]\( k = 7 \)[/tex] into the general form of the equation:
[tex]\[
y = 7x
\][/tex]
Therefore, the equation that represents the direct variation function is:
[tex]\[
y = 14x
\][/tex]
From the provided options, the correct one is:
[tex]\[
y = 14x
\][/tex]
So, the answer is:
[tex]\[ \boxed{4} \][/tex]
