qanda-logo
search-icon
Symbol

Calculator search results

Convert to the standard form of the quadratic function
Answer
circle-check-icon
expand-arrow-icon
Find the maximum and minimum of the quadratic function
Answer
circle-check-icon
expand-arrow-icon
Graph
$y = - x ^ { 2 } - x - 2$
$y$Intercept
$\left ( 0 , - 2 \right )$
$y = - \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } - \dfrac { 7 } { 4 }$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } - 2$
$ $ In order to convert the quadratic equation on the right side to the standard form, enclose it with the coefficient of the highest term $ $
$y = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \right ) - 2$
$y = - \left ( x ^ { 2 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } x \right ) \right ) - 2$
$ $ Simplify Minus $ $
$y = - \left ( x ^ { 2 } + x \right ) - 2$
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \right ) - 2$
$ $ Add and subtract constants to convert the quadratic equation on the right side to the standard form $ $
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) - 2$
$y = - \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \dfrac { 1 } { 2 } \right ) ^ { 2 } \right ) - 2$
$ $ Organize the expression using $ A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = - \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \dfrac { 1 } { 2 } \right ) ^ { 2 } \right ) - 2$
$y = - \left ( \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } - \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) - 2$
$ $ Calculate power $ $
$y = - \left ( \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } - \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) - 2$
$y = \color{#FF6800}{ - } \left ( \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) - 2$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$y = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } - 2$
$y = - \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$ $ Subtract $ 2 $ from $ \dfrac { 1 } { 4 }$
$y = - \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 4 } } }$
$- \dfrac { 7 } { 4 }$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$ $ Rewrite it as the standard form of the quadratic function $ $
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 4 } } }$
$\color{#FF6800}{ y } = \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 4 } } }$
$ $ As $ a \lt 0 $ is, the maximum value is $ - \dfrac { 7 } { 4 } $ if $ x $ = $ - \dfrac { 1 } { 2 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 4 } } }$
Have you found the solution you wanted?
Try again
Try more features at Qanda!
check-iconSearch by problem image
check-iconAsk 1:1 question to TOP class teachers
check-iconAI recommend problems and video lecture