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Formula
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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Calculate the differentiation
Answer
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Graph
$y = x ^ { 2 } - 4 x - 1$
$x$-intercept
$\left ( 2 - \sqrt{ 5 } , 0 \right )$, $\left ( 2 + \sqrt{ 5 } , 0 \right )$
$y$-intercept
$\left ( 0 , - 1 \right )$
Minimum
$\left ( 2 , - 5 \right )$
Standard form
$y = \left ( x - 2 \right ) ^ { 2 } - 5$
$y = x ^{ 2 } -4x-1$
$y = \left ( x - 2 \right ) ^ { 2 } - 5$
Rewrite it as the standard form of the quadratic function
$y = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Add and subtract constants to convert the quadratic equation on the right side to the standard form $ $
$y = x ^ { 2 } - 4 x - 1 + \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$
$y = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 1 \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 2 ^ { 2 }$
$ $ Organize the expression using $ A^{2} ± 2AB + B^2 = (A ± B)^{2}$
$y = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } - 1 - 2 ^ { 2 }$
$y = \left ( x - 2 \right ) ^ { 2 } - 1 - \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$
$ $ Calculate power $ $
$y = \left ( x - 2 \right ) ^ { 2 } - 1 - \color{#FF6800}{ 4 }$
$y = \left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ Find the sum of the negative numbers $ $
$y = \left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 5 }$
$- 5$
Find the maximum and minimum of the quadratic function
$\color{#FF6800}{ y } = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Rewrite it as the standard form of the quadratic function $ $
$\color{#FF6800}{ y } = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 }$
$\color{#FF6800}{ y } = \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 }$
$ $ As $ a \gt 0 $ is, the minimum value is $ - 5 $ if $ x = 2$
$\color{#FF6800}{ - } \color{#FF6800}{ 5 }$
$\dfrac {d } {d x } {\left( y \right)} = 2 x - 4$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right)}$
$ $ Calculate the differentiation $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ 그래프 보기 $ $
Quadratic function
Solution search results
search-thumbnail-$1f$ $fx=1$ 
$y=x^{2}-4x-5$ 
$y=1^{2}-4\left(1\right)-5$
7th-9th grade
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