Answer:
1) y = (1/15)x + 18
2) y = (5/144) (x - 16)^2 + 20
Solution:
1) The solution consists in establishing the equations that model the two paths.
2) Let's start with the easiest one. It is the linear path.
You have that the equation is a line that passes through the points (0,18) and (30,20). This equation is found using this formula:
y - b d - b
------ = ----------
x - a c - a
where the two points are (a,b) and (c,d)
So, for the two points given:
y - 18 20 - 18 2
-------- = ----------- = ------
x - 0 30 - 0 30
=> (y - 18)*30 = x*2
=>30y - 540 = 2x
=> 30y = 2x + 540
=> y = 2x / 30 + 540 / 30
=> y = x/15 + 18 ---------------> this is the first equation of the system.
3) Now, find the equation for the path modeled by the quadratic function.
The vertex form of a quadratic function is: y = A (x - h)^2 + k
where the vertex is (k,h).
Then, so far you can write y = A (x - 16)^2 + 20
Now, replace the coordinates of the point (4,25) to find the value of A:
25 = A (4 - 16)^2 + 20
=> 25 = A (-12)^2 + 20
=> 25 = A*144 + 20
=> 144A = 25 - 20
=> 144 A = 5
=> A = 5 / 144
=> y = (5 / 144) (x - 16)^2 + 20 ---------> this is the second equation of the system.
System:
1) y = (1/15)x + 18
2) y = (5/144) (x - 16)^2 + 20
