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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
$y = x ^ { 2 } - 6 x + 7$
$x$Intercept
$\left ( 3 - \sqrt{ 2 } , 0 \right )$, $\left ( \sqrt{ 2 } + 3 , 0 \right )$
$y$Intercept
$\left ( 0 , 7 \right )$
Minimum
$\left ( 3 , - 2 \right )$
Standard form
$y = \left ( x - 3 \right ) ^ { 2 } - 2$
$y = x ^{ 2 } -6x+7$
$y = \left ( x - 3 \right ) ^ { 2 } - 2$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 }$
 Add and subtract constants to convert the quadratic equation on the right side to the standard form 
$y = x ^ { 2 } - 6 x + 7 + \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } }$
$y = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } + 7 \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } - 3 ^ { 2 }$
 Organize the expression using $A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } + 7 - 3 ^ { 2 }$
$y = \left ( x - 3 \right ) ^ { 2 } + 7 - \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } }$
 Calculate power 
$y = \left ( x - 3 \right ) ^ { 2 } + 7 - \color{#FF6800}{ 9 }$
$y = \left ( x - 3 \right ) ^ { 2 } + \color{#FF6800}{ 7 } \color{#FF6800}{ - } \color{#FF6800}{ 9 }$
 Subtract $9$ from $7$
$y = \left ( x - 3 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$- 2$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 }$
 Rewrite it as the standard form of the quadratic function 
$\color{#FF6800}{ y } = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$\color{#FF6800}{ y } = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 As $a \gt 0$ is, the minimum value is $- 2$ if $x = 3$
$\color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$\dfrac {d } {d x } {\left( y \right)} = 2 x - 6$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 } \right)}$
 Calculate the differentiation 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 6 }$
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