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Math Help - Finding Equation of a Parabola from Tangent Slopes

  1. #1
    Senior Member topher0805's Avatar
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    Finding Equation of a Parabola from Tangent Slopes

    Find a parabola of the form given below that has slope m_1 at x_1, slope m_2 at x_2, and passes through the point P.

    <br />
y = ax^2 + bx + c

    m_1 = 9, x_1 = 10

    m_2 = 4, x_2 = 6

    P = (9, 2)
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  2. #2
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    Hello, topher0805!

    Find a parabola of the form y \:=\:ax^2+bx+c .that has slope 9 at x=10,
    slope 4 at x=6, and passes through the point (9,2)
    The slope is given by: . y' \:=\:2ax + b


    \begin{array}{cccccc}<br />
x=10,\:y'=9\!: & 2a(10) + b \:=\:9 & \Rightarrow & 20a + b \:=\:9 & {\color{blue}[1]}\\<br />
x=6,\:y' = 4\!: & 2a(6) + b \:=\:4 & \Rightarrow &  12a + b \:=\:4 & {\color{blue}[2]} \\<br />
(9,2)\!: &&& 81a + 9b + c \:=\:2 & {\color{blue}[3]}\end{array}


    Subtract [2] from [1]: . 8a\:=\:5\quad\Rightarrow\quad\boxed{ a \:=\:\frac{5}{8}}

    Substitute into [2]: . 12\left(\frac{5}{8}\right) + b \:=\:4\quad\Rightarrow\quad\boxed{ b \:=\:-\frac{7}{2}}

    Substitute into [3]: . 81\left(\frac{5}{8}\right) + 9\left(-\frac{7}{2}\right) + c \:=\:2\quad\Rightarrow\quad\boxed{ c \:=\:-\frac{137}{8}}


    Therefore: . \boxed{y \;=\;\frac{5}{8}x^2 - \frac{7}{2}x - \frac{137}{8}}

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