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Convert to the standard form of the quadratic function
Answer
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Find the maximum and minimum of the quadratic function
Answer
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Calculate the differentiation
Answer
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Graph
$y = - 2 x ^ { 2 } + 4 x - 1$
$x$Intercept
$\left ( \dfrac { \sqrt{ 2 } } { 2 } + 1 , 0 \right )$, $\left ( 1 - \dfrac { \sqrt{ 2 } } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , - 1 \right )$
Maximum
$\left ( 1 , 1 \right )$
Standard form
$y = - 2 \left ( x - 1 \right ) ^ { 2 } + 1$
$y = -2x ^{ 2 } +4x-1$
$y = - 2 \left ( x - 1 \right ) ^ { 2 } + 1$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 1$
$ $ In order to convert the quadratic equation on the right side to the standard form, enclose it with the coefficient of the highest term $ $
$y = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) - 1$
$y = - 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) - 1$
$ $ Add and subtract constants to convert the quadratic equation on the right side to the standard form $ $
$y = - 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \right ) - 1$
$y = - 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - 1 ^ { 2 } \right ) - 1$
$ $ Organize the expression using $ A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = - 2 \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } - 1 ^ { 2 } \right ) - 1$
$y = - 2 \left ( \left ( x - 1 \right ) ^ { 2 } - \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \right ) - 1$
$ $ Calculate power $ $
$y = - 2 \left ( \left ( x - 1 \right ) ^ { 2 } - \color{#FF6800}{ 1 } \right ) - 1$
$y = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) - 1$
$ $ Multiply each term in parentheses by $ - 2$
$y = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } + \color{#FF6800}{ 2 } - 1$
$y = - 2 \left ( x - 1 \right ) ^ { 2 } + \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Subtract $ 1 $ from $ 2$
$y = - 2 \left ( x - 1 \right ) ^ { 2 } + \color{#FF6800}{ 1 }$
$1$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Rewrite it as the standard form of the quadratic function $ $
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
$ $ As $ a \lt 0 $ is, the maximum value is $ 1 $ if $ x $ = $ 1$
$\color{#FF6800}{ 1 }$
$\dfrac {d } {d x } {\left( y \right)} = - 4 x + 4$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right)}$
$ $ Calculate the differentiation $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 4 }$
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