Certainly! Let's identify the equation in the point-slope form for the line perpendicular to [tex]\( y = \frac{1}{4} x - 7 \)[/tex] that passes through the point [tex]\((-2, -6)\)[/tex].
1. Determine the slope of the given line:
The given line is [tex]\( y = \frac{1}{4} x - 7 \)[/tex]. The slope (m) of this line is [tex]\( \frac{1}{4} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. For the original slope [tex]\( \frac{1}{4} \)[/tex], the negative reciprocal is:
[tex]\[
-\frac{1}{ \left( \frac{1}{4} \right) } = -4
\][/tex]
3. Use the point-slope form of the equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. We have [tex]\((-2, -6)\)[/tex] as the point and [tex]\(-4\)[/tex] as the slope. Substituting these into the point-slope form:
[tex]\[
y - (-6) = -4(x - (-2))
\][/tex]
Simplifying:
[tex]\[
y + 6 = -4(x + 2)
\][/tex]
Thus, the equation in point-slope form for the line perpendicular to [tex]\( y = \frac{1}{4} x - 7 \)[/tex] that passes through the point [tex]\((-2, -6)\)[/tex] is:
[tex]\[
y + 6 = -4(x + 2)
\][/tex]
This corresponds to option A:
[tex]\[
\boxed{y + 6 = -4(x + 2)}
\][/tex]
