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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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Graph
$y = - x ^ { 2 } - 4 x - 4$
$x$Intercept
$\left ( - 2 , 0 \right )$
$y$Intercept
$\left ( 0 , - 4 \right )$
$y = - \left ( x + 2 \right ) ^ { 2 }$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 4$
$ $ In order to convert the quadratic equation on the right side to the standard form, enclose it with the coefficient of the highest term $ $
$y = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) \color{#FF6800}{ x } \right ) - 4$
$y = - \left ( x ^ { 2 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 4 \right ) x \right ) - 4$
$ $ Simplify Minus $ $
$y = - \left ( x ^ { 2 } + 4 x \right ) - 4$
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \right ) - 4$
$ $ Add and subtract constants to convert the quadratic equation on the right side to the standard form $ $
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) - 4$
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) - 4$
$ $ Organize the expression using $ A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = - \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 } \right ) - 4$
$y = - \left ( \left ( x + 2 \right ) ^ { 2 } - \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right ) - 4$
$ $ Calculate power $ $
$y = - \left ( \left ( x + 2 \right ) ^ { 2 } - \color{#FF6800}{ 4 } \right ) - 4$
$y = \color{#FF6800}{ - } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) - 4$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$y = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } + \color{#FF6800}{ 4 } - 4$
$y = - \left ( x + 2 \right ) ^ { 2 } + \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ Remove the two numbers if the values are the same and the signs are different $ $
$y = - \left ( x + 2 \right ) ^ { 2 } + 0$
$y = - \left ( x + 2 \right ) ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 0 }$
$ $ 0 does not change when you add or subtract $ $
$y = - \left ( x + 2 \right ) ^ { 2 }$
$0$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ Rewrite it as the standard form of the quadratic function $ $
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ As $ a \lt 0 $ is, the maximum value is $ 0 $ if $ x $ = $ - 2$
$\color{#FF6800}{ 0 }$
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