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Math Help - Graph Intersection

  1. #1
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    Graph Intersection

    Considering the graphs of y = 3x + c and y^2 = 6x, where c is a real constant, determine all values of c for which the graphs intersect in two distinct points.

    I'm not entirely sure of where to start so any help is greatly appreciated.
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  2. #2
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    Hello, Naples!

    Considering the graphs of: . \begin{Bmatrix}y \:=\: 3x + c \\ y^2 \:=\: 6x\end{Bmatrix} .where c is a real constant,

    determine all values of c for which the graphs intersect in two distinct points.
    First, find the intersections . . .

    The first equation gives us: . x \:=\:\frac{y-c}{3}

    Substitute into the second: . y^2 \:=\:6\left(\frac{y-c}{3}\right) \quad\Rightarrow\quad y^2 - 2y + 2c \:=\:0

    Quadratic Formula: . y \;=\;\frac{2\pm\sqrt{4-8c}}{2}

    The quadratic has two roots if the discriminant is positive: . 4-8c \:>\:0

    Therefore: . -8c \:>\:-4 \quad\Rightarrow\quad\boxed{ c \:<\:\tfrac{1}{2}}

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