Two boats depart from a port located at (–8, 1) in a coordinate system measured in kilometers and travel in a positive x-direction. the first boat follows a path that can be modeled by a quadratic function with a vertex at (1, 10), whereas the second boat follows a path that can be modeled by a quadratic function with a vertex at (0, –7). which system of equations can be used to determine whether the paths of the boats cross?

For a quadratic equation ax² + bx + c = 0

Both boats start from (-8, 1)

Vertex of first boat = (1, 10)

Vertex of second boat = (0, -7)

1st boat  has common points  (-8, 1) and (1, 10)

2nd boat  has common points  (-8, 1) and (0, -7)

Let the different quadratic equations for both boats be:

 y₁ = a₁x² + b₁x + c₁  and  y₂ = a₂x² + b₂x + c₂

So for 1st equation.

 y₁ = a₁x² + b₁x + c₁      for  (-8, 1)

1 = a*(-8)² + b*(-8) + c 

1 = 64a₁ - 8b₁ + c₁

64a₁ - 8b₁ + c₁ = 1(a)

For point (1, 10)

y₁ = a₁x² + b₁x + c₁      for  (1, 10)

10 = a₁*(1)² + b₁*(1) + c₁ 

10 = a₁ + b₁ + c₁

a₁ + b₁ + c₁ = 10(b)

So for 2nd equation.

 y₂ = a₂x² + b₂x + c₂      for  (-8, 1)

1 = a₂*(-8)² + b₂*(-8) + c₂ 

1 = 64a₂ - 8b₂ + c₂

64a₂ - 8b₂ + c₂ = 1(c)

for second point  (0, -7)

-7 = a₂*(0)² + b₂*(0) + c₂ 

-7 = 0 - 0 + c₂

c₂ = -7 (d)

So system of equations are:

64a₁ - 8b₁ + c₁ = 1(a)

a₁ + b₁ + c₁ = 10(b)

64a₂ - 8b₂ + c₂ = 1  (c)

c₂ = -7                (d)

So these are the systems of equations. When  y₁ = a₁x² + b₁x + c₁  and  y₂ = a₂x² + b₂x + c₂ are solved and there is a solution other than the first initial point of (-8, 1) then it means the paths crossed and if there no other solution than (-8, 1) it means their paths did not cross.  

Lines that intersect at one point