Question

Functions and graphs

Answer 1

f(x) = -8 x + 34

Explanation:

First, normalize the equation 8y = x-16 into y = ... form (divide by 8):

y = 1/8 x - 2

So the slope of this line is 1/8. The slope of a perpendicular line is the negative reciprocal, which means you swap numerator and denominator and add a minus sign. So the slope of our line is -8 and our function will look like f(x) = -8x + b

Now all we have to do is find b such that f(x) goes through (5,-6).

So f(5) = -8*5 + b = -6, and solve it for b:

-40 + b = -6 =>

b = 34.

So f(x) = -8x + 34

Answer 2

`\text{Let}\ k:y=m_1x+b_1\ \text{and}\ l:y=m_2x+b_2.\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1).\\\\\text{We have}\ 8y=x-16\qquad\text{divide both sides by 8}\\\\y=(1)/(8)x-2\to m_1=(1)/(8).\\\\\text{Let}\ y=mx+b.\ \text{It's perpendicular to given line. Therefore}\ m=-(1)/((1)/(8))=-8.\\\\y=-8x+b.\\\\\text{The line passes through point (5, -6)}.\ \text{Put the coordinates of the point}\\\text{to the equation of a line:}`

`-6=-8(5)+b\\-6=-40+b\qquad\text{add 40 to both sides}\\34=b\to b=34\\\\Answer:\ \boxed{f(x)=-8x+34}`