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Formula
Convert to the standard form of the quadratic function
Find the maximum and minimum of the quadratic function
Calculate the differentiation
Graph
$y = 2 x ^ { 2 } - 8 x + 5$
$x$Intercept
$\left ( \dfrac { \sqrt{ 6 } } { 2 } + 2 , 0 \right )$, $\left ( 2 - \dfrac { \sqrt{ 6 } } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , 5 \right )$
Minimum
$\left ( 2 , - 3 \right )$
Standard form
$y = 2 \left ( x - 2 \right ) ^ { 2 } - 3$
$y = 2x ^{ 2 } -8x+5$
$y = 2 \left ( x - 2 \right ) ^ { 2 } - 3$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } + 5$
 In order to convert the quadratic equation on the right side to the standard form, enclose it with the coefficient of the highest term 
$y = \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \right ) + 5$
$y = 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \right ) + 5$
 Add and subtract constants to convert the quadratic equation on the right side to the standard form 
$y = 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) + 5$
$y = 2 \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) + 5$
 Organize the expression using $A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = 2 \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) + 5$
$y = 2 \left ( \left ( x - 2 \right ) ^ { 2 } - \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) + 5$
 Calculate power 
$y = 2 \left ( \left ( x - 2 \right ) ^ { 2 } - \color{#FF6800}{ 4 } \right ) + 5$
$y = \color{#FF6800}{ 2 } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 5$
 Multiply each term in parentheses by $2$
$y = \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } + \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 5$
$y = 2 \left ( x - 2 \right ) ^ { 2 } + \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 5$
 Multiply $2$ and $- 4$
$y = 2 \left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } + 5$
$y = 2 \left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 5 }$
 Add $- 8$ and $5$
$y = 2 \left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$- 3$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 }$
 Rewrite it as the standard form of the quadratic function 
$\color{#FF6800}{ y } = \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ y } = \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 As $a \gt 0$ is, the minimum value is $- 3$ if $x = 2$
$\color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\dfrac {d } {d x } {\left( y \right)} = 4 x - 8$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right)}$
 Calculate the differentiation 
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 8 }$
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