How do I know if, for example, a line goes through the points (3,15) and (5,6)? How do I know if there's an exponential function that goes through these two points?

Answer 1

If you are given the equation for the line, you can substitute the values of x and y into the equation. If the equation checks out, meaning the values make the equation correct, that coordinate is on the line.

Answer 2

(3,15) and (5,6) have only one unique straight line through them

but an infinite number of curved lines go through the 2 points too

straight line is y-6 = (9/26)(x-5)

curved lines include an exponential function

where y=ab^x + c, plug in the points and get 2 equations

15 = ab^3 + c

6 = ab^5 +c

15-c = ab^3

6-c = ab^5

divide 2nd to last into last equation, to last to eliminate one variable, a:

(6-c)/(15-c) = b^2

let c=0, then exponential function is y = 37.5sqr10(.2sqr10)^x

let c = any other number and there are an infinite number of possible exponential equations through the 2 given points