Questions: A direct variation function contains the points (-8,-6) and (12,9). Which equation represents the function? y=-4/3 x y=-3/4 x y=3/4 x y=4/3 x

A direct variation function contains the points (-8,-6) and (12,9). Which equation represents the function?
y=-4/3 x
y=-3/4 x
y=3/4 x
y=4/3 x
Transcript text: A direct variation function contains the points $(-8,-6)$ and $(12,9)$. Which equation represents the function? $y=-\frac{4}{3} x$ $y=-\frac{3}{4} x$ $y=\frac{3}{4} x$ $y=\frac{4}{3} x$
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Solution

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Solution Steps

To find the equation of a direct variation function given two points, we need to determine the slope of the line that passes through these points. The slope is calculated as the change in y divided by the change in x. Once we have the slope, we can write the equation in the form y=mx y = mx , where m m is the slope.

Step 1: Calculate the Slope

To find the slope m m of the line that passes through the points (8,6) (-8, -6) and (12,9) (12, 9) , we use the formula:

m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the coordinates of the points:

m=9(6)12(8)=9+612+8=1520=34 m = \frac{9 - (-6)}{12 - (-8)} = \frac{9 + 6}{12 + 8} = \frac{15}{20} = \frac{3}{4}

Step 2: Write the Equation

The equation of a direct variation function can be expressed in the form:

y=mx y = mx

Substituting the calculated slope m=34 m = \frac{3}{4} :

y=34x y = \frac{3}{4}x

Final Answer

The equation that represents the function is \\(\boxed{y = \frac{3}{4} x}\\).

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