Question

If k is a non-zero constant, determine the vertex of the function y = x2 – 2kx + 3 in terms of k.

Answers

  2. Answer:

    \displaystyle \text{Vertex} = \left(k, 3-k^2\right)

    Step-by-step explanation:

    We are given the quadratic equation:

    y=x^2-2kx+3

    Where k is a non-zero constant.

    And we want to determine the vertex of the parabola in terms of k.

    The vertex of a parabola is given by the formulas:

    \displaystyle \text{Vertex}=\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)

    In this case, a = 1, b = -2k, and c = 3.

    Find the x-coorinate of the vertex:

    \displaystyle x=-\frac{(-2k)}{2(1)}=\frac{2k}{2}=k

    To find the y-coordinate, we substitute the value we acquired back into the equation. So:

    \displaystyle \begin{aligned} y(k)&=(k)^2-2k(k)+3\\&=k^2-2k^2+3\\&=3-k^2\end{aligned}

    Therefore, our vertex in terms of k is:

    \displaystyle \text{Vertex} = \left(k,3-k^2\right)

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