Differential Equations-Finding Constants

[Bashyboy]

Differential Equations--Finding Constants

Homework Statement

I uploaded a pdf document, which contains the problem I am currently working on, namely, problem number 7

Homework Equations

The Attempt at a Solution

I am having particular difficulty with this portion of the solution:

"(ii) At y=3, the function ay+b changes from positive to negative
--> ay+b has a negative slope --> a is negative"

Why does "a" being negative follow from these facts? Also, why does a function change signs at y = 3?

 

[voko]

Two lines above, it was shown that ay + b > 0 when y < 3, and ay + b < 0 when y > 3. That literally means that it changes its sign at y = 3. And because it goes from positive to negative, it has to have a negative slope. Now this function is a simple straight line, so its slope = a, hence a must be negative.