To find the point-slope form of the equation of the line that contains the point [tex]\((-2, 12)\)[/tex] and has a given function [tex]\( f(x) = -5x + 2 \)[/tex], follow these steps:
1. Identify the slope ([tex]\(m\)[/tex]):
The given function is [tex]\( f(x) = -5x + 2 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex].
2. Identify the point [tex]\((x_1, y_1)\)[/tex] through which the line passes:
The point provided is [tex]\((-2, 12)\)[/tex], so we have [tex]\( x_1 = -2 \)[/tex] and [tex]\( y_1 = 12 \)[/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]
4. Substitute the known values into the point-slope form:
Substituting the slope [tex]\( m = -5 \)[/tex] and the point [tex]\((-2, 12)\)[/tex] into the equation:
[tex]\[
y - 12 = -5(x - (-2))
\][/tex]
5. Simplify the equation:
[tex]\[
y - 12 = -5(x + 2)
\][/tex]
So, the point-slope form of the equation of the line that Mr. Shaw graphed is:
[tex]\[
y - 12 = -5(x + 2)
\][/tex]
Thus, the correct answer is:
[tex]\[
y - 12 = -5(x + 2)
\][/tex]
