Respuesta :

bearing in mind that perpendicular lines have negative reciprocal slopes, hmmm what is the slope of that equation in the graph anyway?

well, notice, the equation is in slope-intercept form, thus    [tex]\bf y=\stackrel{slope}{\cfrac{1}{4}}x-7[/tex]

hmm what is the slope of a perpendicular to that then?

[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{\cfrac{1}{4}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{4}{1}}\qquad \stackrel{negative~reciprocal}{-\cfrac{4}{1}}}\implies -4[/tex]

so, we're really looking for the equation of a line whose slope is -4 and runs through -2,-6,

[tex]\bf (\stackrel{x_1}{-2}~,~\stackrel{y_1}{-6})\qquad \qquad \qquad slope = m\implies -4 \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-(-6)=-4[x-(-2)] \\\\\\ y+6=-4(x+2)[/tex]

The equation y + 6 = -4 (x+2) at option B is the required equation in point-slope form for the line perpendicular to y = 1/4x - 7 that passes through (-2, -6).

What is the point-slope form of a line?

The point-slope form of a line having slope m and passing through the point (x1, y1) is (y - y1) = m (x - x1).

Calculating the perpendicular line equation:

The given equation of the line is y = 1/4x - 7 and a point (-2, -6)

The slope of the line y = 1/4x - 7 is m = 1/4

So, the slope of its perpendicular line is -1/m

⇒ [tex]\frac{-1}{\frac{1}{4} }[/tex]

⇒ -4

Then the equation of a line with point (-2, -6) and slope -4 is calculated as follows:

(y - y1) = m (x - x1)

⇒ (y - (-6)) = -4(x - (-2))

⇒ (y + 6) = -4(x + 2)

Thus, the equation at option B represents the equation of the perpendicular line.

Learn more about point-slope form here:

https://brainly.com/question/2019733

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