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MAT 101 Online Stewart Section 4.1: Quadratic Functions and Models

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  • Started 2 years ago by lee

  1. lee
    Key Master

    In Stewart's textbook, you are instructed to rewrite a given quadratic function $y=ax^2+bx+c$ in the standard form $y=a(x-k)^2+h$ by completing the square. However, it is unnecessary. All you have to know is the formula for the $x$-coordinate of the vertex $(k,h)$ which is $k=-\frac{b}{2a}$. Note that this formula actually comes from completing the square. The $y$-coordinate of the vertex is then found by calculating $h=f(k)$.

    Example. Express $f(x)=2x^2-12x+23$ in standard form. (Example 1 on page 189.)

    Solution. $k=-\frac{b}{2a}=-\frac{-12}{2\cdot 2}=3$ and
    \begin{align*}
    h&=f(k)\\
    &=f(3)\\
    &=2(3)^2-12(3)+23\\
    &=5.
    \end{align*}
    Hence, in standard form $f(x)=2(x-3)^2+5$.

    Posted 2 years ago #

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