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A direct variation function contains the points [tex]\((-8,-6)\)[/tex] and [tex]\((12,9)\)[/tex]. Which equation represents the function?

A. [tex]\( y = -\frac{4}{3} x \)[/tex]
B. [tex]\( y = -\frac{3}{4} x \)[/tex]
C. [tex]\( y = \frac{3}{4} x \)[/tex]
D. [tex]\( y = \frac{4}{3} x \)[/tex]


Sagot :

To determine which equation represents the direct variation function that contains the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex], we need to find the slope [tex]\( m \)[/tex] of the line that passes through these points.

1. We start by using the formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

2. Substituting the given points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex] into the slope formula:

[tex]\[ m = \frac{9 - (-6)}{12 - (-8)} \][/tex]

3. Simplify the numerator and the denominator:

[tex]\[ m = \frac{9 + 6}{12 + 8} = \frac{15}{20} \][/tex]

4. Simplify the fraction:

[tex]\[ m = \frac{15}{20} = \frac{3}{4} \][/tex]

5. Since a direct variation function has the form [tex]\( y = mx \)[/tex], where [tex]\( m \)[/tex] is the slope we just calculated, we now know that the equation representing the function is:

[tex]\[ y = \frac{3}{4} x \][/tex]

Among the provided choices, the correct equation is:
[tex]\[ y = \frac{3}{4} x \][/tex]

Therefore, the correct choice is:
[tex]\[ y = \frac{3}{4} x \][/tex]
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