Question

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 1

The answer is A

Explanation:

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Answer 2

`\left\{\begin{array}{l}y=-(1)/(9)x^2 +(2)/(9)x+(89)/(9)\\ \\y=(1)/(8)x^2 -7\end{array}\right.`

Explanation:

1st boat:

Parabola equation:

`y=ax^2 +bx+c`

The x-coordinate of the vertex:

`x_v=-(b)/(2a)\Rightarrow -(b)/(2a)=1\\ \\b=-2a`

Equation:

`y=ax^2 -2ax+c`

The y-coordinate of the vertex:

`y_v=a\cdot 1^2-2a\cdot 1+c\Rightarrow a-2a+c=10\\ \\c-a=10`

Parabola passes through the point (-8,1), so

`1=a\cdot (-8)^2-2a\cdot (-8)+c\\ \\80a+c=1`

Solve:

`c=10+a\\ \\80a+10+a=1\\ \\81a=-9\\ \\a=-(1)/(9)\\ \\b=-2a=(2)/(9)\\ \\c=10-(1)/(9)=(89)/(9)`

Parabola equation:

`y=-(1)/(9)x^2 +(2)/(9)x+(89)/(9)`

2nd boat:

Parabola equation:

`y=ax^2 +bx+c`

The x-coordinate of the vertex:

`x_v=-(b)/(2a)\Rightarrow -(b)/(2a)=0\\ \\b=0`

Equation:

`y=ax^2+c`

The y-coordinate of the vertex:

`y_v=a\cdot 0^2+c\Rightarrow c=-7`

Parabola passes through the point (-8,1), so

`1=a\cdot (-8)^2-7\\ \\64a-7=1`

Solve:

`a=-(1)/(8)\\ \\b=0\\ \\c=-7`

Parabola equation:

`y=(1)/(8)x^2 -7`

System of two equations:

`\left\{\begin{array}{l}y=-(1)/(9)x^2 +(2)/(9)x+(89)/(9)\\ \\y=(1)/(8)x^2 -7\end{array}\right.`