Question

Math Latex Test

Answer 1

\(\bbox[6pt,#9FFFFF ,border:8px solid black]{\Large\left.\begin{matrix} ~~ & ~~ & 1 & 2 & 3 & 4 & 5 & 6 \\
~~ &~~ & \text{___} & \text{___} & \text{___} & \text{___} & \text{___} & \text{___} \\[0.5em]
~~1 & | & \color{red}{2} & \color{red}{3} & \color{red}{4} & \color{red}{5} & \color{red}{6} & \color{red}{7} \\[0.5em]
~~2 & | & \color{red}{3} & \color{red}{4} & \color{red}{5} & \color{red}{6} & \color{red}{7} & 8 \\[0.5em]
~~3 & | & \color{red}{4} & \color{red}{5} & \color{red}{6} & \color{red}{7} & 8 & 9 \\[0.5em]
~~4 & | & \color{red}{5} & \color{red}{6} & \color{red}{7} & 8 & 9 & 10 \\[0.5em]
~~5 & | & \color{red}{6} & \color{red}{7} & 8 & 9 & 10 & 11 \\[0.5em]
~~6 & | & \color{red}{7} & 8 & 9 & 10 & 11 & 12 \end{matrix}\right. }\)

Answer 2

\(\normalsize\color{ slate }{{\bbox[5pt, lightcyan ,border:2px solid black ]{ \LARGE{x=~} \huge{ \frac{-\color{magenta}{b} \pm\sqrt{ \color{magenta}{~b} ^2-4 \color{blue}{a} \color{red}{c}~}}{2 \color{blue}{a}} }~ }}}\)

when the equation is \(\large\color{black}{ \color{blue}{a} x^2+ \color{magenta}{b}x+ \color{red}{c}=0 }\).

Answer 3

\(\bbox[8pt, #99ff99,border:1px solid black]{\Large \Large{x=~} \huge{ \frac{-\color{magenta}{b} \pm\sqrt{ \color{magenta}{b} ^2-4 \color{blue}{a} \color{red}{c}}}{2 \color{blue}{a}} } ~ }\)

when the equation is \(\Large\color{black}{ \color{blue}{a} x^2+ \color{magenta}{b}x+ \color{red}{c}=0 }\).

Answer 4

Rules of \(\large\color{black}{ \rm shifts }\) from \(\large\color{black}{ \rm f(x) }\) to \(\large\color{black}{ \rm g(x) }\).

\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \color{blue}{ -~\rm{c} }\right| }\)
\(\large\color{blue}{ ~\rm {c} }\) units to the \(\normalsize\color{blue}{ \rm right }\).

\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \color{blue}{ +~\rm{c} }\right| }\)
\(\large\color{blue}{ ~\rm {c} }\) units to the \(\normalsize\color{blue}{ \rm left }\).

\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \right| \color{blue}{ +~\rm{c} }}\)
\(\large\color{blue}{ ~\rm {c} }\) units \(\normalsize\color{blue}{ \rm up }\).

\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \right| \color{blue}{ -~\rm{c} }}\)
\(\large\color{blue}{ ~\rm{c} }\) units \(\normalsize\color{blue}{ \rm down }\).

Also,
the ` reflection across the X -axis. `

\(\large\color{red}{ \rm f(x)=\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=\color{blue}{ - }\left| x \right| }\)
(the \(\large\color{red}{ ~\rm{f(x)} }\) and \(\normalsize\color{red}{ \rm g(x) }\) are mirrors of each other over the \(\large\color{red}{ \rm{x-axis} }\). ) \(\LARGE\color{white}{ \rm │ }\)

And lastly,
\(\normalsize\color{black}{ \rm{ s~t~r~e~t~c~h~i~n~g} }\)

\(\large\color{black}{ \rm f(x)=\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=\color{blue}{ c }\left| x \right| }\)

For any real number \(\normalsize\color{blue}{ \rm{c} }\), (provided that \(\normalsize\color{blue}{ \rm{c\neq1~~or~~0} }\) )

\(\normalsize\color{black}{ \rm{1)} }\) When \(\normalsize\color{blue}{ \rm{\left| c \right| >1} }\) the (new function) \(\normalsize\color{black}{ \rm{g(x)} }\) is streched \(\normalsize\color{blue}{ \rm{ vertically} }\).
(if comparing to the initial function \(\normalsize\color{black}{ \rm{f(x)} }\). )
\(\normalsize\color{black}{ \rm{2)} }\) When \(\normalsize\color{blue}{ \rm{\left| c \right| <1} }\) the (new function) \(\normalsize\color{black}{ \rm{g(x)} }\) is streched \(\normalsize\color{blue}{ \rm{ horizontally} }\).
(if comparing to the initial function \(\normalsize\color{black}{ \rm{f(x)} }\). )

Answer 5

\(\large\color{#000000}{ \displaystyle \left.\int_{a}^{b} f(x)={\rm F}(x)\right|^{b}_{a} ={\rm F}(b)-{\rm F}(a) }\)

Answer 6

\(\Large\color{#993333}{\unicode{ x372 }}\) \(\Large\color{#993333}{\unicode{ x393 }}\) \(\Large\color{#993333}{\unicode{ x394 }}\) \(\Large\color{#993333}{\unicode{ x398 }}\) \(\Large\color{#993333}{\unicode{ x461 }}\)
\(\Large\color{#993333}{\unicode{ x470 }}\) \(\Large\color{#993333}{\unicode{ x472 }}\) \(\Large\color{#993333}{\unicode{ x26E8 }}\) \(\Large\color{#993333}{\unicode{ x146d }}\) \(\Large\color{#993333}{\unicode{ x28d }}\)
\(\Large\color{#993333}{\unicode{ x25F4 }}\) \(\Large\color{#993333}{\unicode{ x26F4 }}\) \(\Large\color{#993333}{\unicode{ x26F5 }}\)