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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
$y = \dfrac { 1 } { 2 } x ^ { 2 } + 2 x + 3$
$y$Intercept
$\left ( 0 , 3 \right )$
Minimum
$\left ( - 2 , 1 \right )$
Standard form
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } + 1$
$y = \dfrac{ 1 }{ 2 } x ^{ 2 } +2x+3$
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } + 1$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } + 3$
 In order to convert the quadratic equation on the right side to the standard form, enclose it with the coefficient of the highest term 
$y = \color{#FF6800}{ \dfrac { 1 } { 2 } } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \right ) + 3$
$y = \dfrac { 1 } { 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \right ) + 3$
 Add and subtract constants to convert the quadratic equation on the right side to the standard form 
$y = \dfrac { 1 } { 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) + 3$
$y = \dfrac { 1 } { 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) + 3$
 Organize the expression using $A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = \dfrac { 1 } { 2 } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) + 3$
$y = \dfrac { 1 } { 2 } \left ( \left ( x + 2 \right ) ^ { 2 } - \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) + 3$
 Calculate power 
$y = \dfrac { 1 } { 2 } \left ( \left ( x + 2 \right ) ^ { 2 } - \color{#FF6800}{ 4 } \right ) + 3$
$y = \color{#FF6800}{ \dfrac { 1 } { 2 } } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 3$
 Multiply each term in parentheses by $\dfrac { 1 } { 2 }$
$y = \color{#FF6800}{ \dfrac { 1 } { 2 } } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } + \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 3$
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } + \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) + 3$
 Calculate the product of rational numbers 
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } + 3$
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
 Add $- 2$ and $3$
$y = \dfrac { 1 } { 2 } \left ( x + 2 \right ) ^ { 2 } + \color{#FF6800}{ 1 }$
$1$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
 Rewrite it as the standard form of the quadratic function 
$\color{#FF6800}{ y } = \color{#FF6800}{ \dfrac { 1 } { 2 } } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
$\color{#FF6800}{ y } = \color{#FF6800}{ \dfrac { 1 } { 2 } } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
 As $a \gt 0$ is, the minimum value is $1$ if $x = - 2$
$\color{#FF6800}{ 1 }$
$\dfrac {d } {d x } {\left( y \right)} = x + 2$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right)}$
 Calculate the differentiation 
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$
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