How to deal with functions+lines with an outstanding exampl

Basics:
if a straight line passes through two different points such as (a,b) and (c,d) then,
the slope of the line = b-d/a-c .

if a function is given and a co-ordinate also with one unknown term its possible to evaluate the unknown one. Such as , y=f(x)=x^3-4x+7 an equation of a curve line and (3,p) is a point on this curve.
So, you can put x=3 and y=p here. So, p = 3^3-43+7=22 .

Now lets apply all these concepts with an example which may seem to you difficult but indeed its not at all.


Example: The graph of x = -y^2+2 and the graph of the line K intersect at (0,p) and (1,q). Which one of the following is the smallest possible slope of line K?
a.-2-1 b. -2+1 c. 2-1 d. 2+1 e.2+2

solution:

(0,p) is on the graph of x = -y^2+2 , so we can evaluate p from here.
0 = -p^2+2
or, p^2 = 2
or, p = 2

Again (1,q) is on the graph of x = -y^2+2 , so we can evaluate q from here.
1= - q^2 + 2
or, q = 1.

We need the slope of the line, so using the 1st formula I mentioned we can evaluate it easily but we have to evaluate the smallest one.

Slope = p q/0-1 = 2 ( 1) / -1
But we need the smallest one. And the denominator is a certain minus one so the numerator should be a positive one for getting the smallest.

Finally, Smallest slope = 2 + 1/ -1 = -2 1
The Answer is a.

(For numerator we chose the highest value to get the smallest one because division by -1 made it lowest finally.)
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