Problema Solution
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?
Answer provided by our tutors
The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k,
where (h, k) is the vertex of the parabola, a is constant
For the first boat we have:
one point: (-8, 1) that is f(-8) = 1
vertex: (1, 10) that is h = 1, k = 10
Plug the values into f(x) = a(x - h)^2 + k
f(x) = a(x - 1)^2 + 10
Now we use f(-8) = 1 to find a:
a(- 8 - 1)^2 + 10 = 1
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click here to see all the equation solution steps
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a = - 9/81
The quadratic equation for the first boat is: f(x) = (-1/9)(x - 1)^2 + 10 or y = (-1/9)(x - 1)^2 + 10
For the second boat we have:
one point: (-8, 1) that is f(-8) = 1
vertex: (0, -7) that is h = 0, k = -7
Plug the values into f(x) = a(x - h)^2 + k
f(x) = a(x - 0)^2 + (-7)
Now we use f(-8) = 1 to find a:
a(- 8 - 0)^2 + (-7) = 1
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click here to see all the equation solution steps
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a = 1/8
The quadratic equation for the second boat is: f(x) = (1/8)x ^2 -7 or y = (1/8)x ^2 -7
The system of equations used to determine whether the paths of the boats cross is:
y = (-1/9)(x - 1)^2 + 10
y = (1/8)x ^2 -7
Click here to see the graphs of the paths of the two boats
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