The graph of $f\left(x\right)$ is shown in black.
In Desmos above, translate $f\left(x\right)$ to match the colored graphs and find the equations in terms of $f\left(x\right)$.
purple graph$\; f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
blue graph$\quad f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
green graph$\;\, f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
red graph$\quad f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
The graph of $f\left(x\right)$ is shown in black.
In Desmos above, translate $f\left(x\right)$ to match the colored graphs and find the equations in terms of $f\left(x\right)$.
purple graph$\; f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
blue graph$\quad f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
green graph$\;\, f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
red graph$\quad f\left(x\right)\rightarrow$
$f\left(x\right)$ has been
by vector
$\bigg($
$\bigg)$
The equation of $f\left(x\right)$ above is $f\left(x\right)=\sqrt{x}$.
$$f\left(x\right)=\sqrt{x}\quad\xrightarrow{\text{translated by }\binom{3}{4}}\quad {\color{#6042a6}f\left(x-3\right)+4}={\color{#6042a6}\sqrt{x-3}+4}$$
Find the equations of the other graphs in terms of $x$.
$\color{#2d70b3}{f\left(x-2\right)-6=}$
$\color{#388c46}{f\left(x+5\right)-4=}$
$\color{#c74440}{f\left(x+8\right)=}$
The graph of $f\left(x\right)=e^x$ is translated by the vector $\displaystyle{\binom{-4}{-1}}$ to form $g\left(x\right)$. Find the equation of $g\left(x\right)$.
$g\left(x\right)=$
The graph of $f\left(x\right)=x^2$ is translated by the vector $\displaystyle{\binom{3}{2}}$ to form $g\left(x\right)$. Find the equation of $g\left(x\right)$ in the form $ax^2+bx+c$ where $a,b,c\in \mathbb{Z}$
$g\left(x\right)=$
The graph of $f\left(x\right)=x^2-x$ is translated by the vector $\displaystyle{\binom{3}{2}}$ to form $g\left(x\right)$. Find the equation of $g\left(x\right)$ in the form $ax^2+bx+c$ where $a,b,c\in \mathbb{Z}$
$g\left(x\right)=$
The graph of $f\displaystyle{\left(x\right)=\frac{1}{2x}}$ is translated by the vector $\displaystyle{\binom{-2}{3}}$ to form $g\left(x\right)$. Find the equation of $g\left(x\right)$ in the form $\displaystyle{\frac{ax+b}{cx+d}}$ where $a,b,c,d\in \mathbb{Z}$
$g\left(x\right)=$
The graph of $f\left(x\right)=\log_2{x}$ is translated by the vector $\displaystyle{\binom{4}{3}}$ to form $g\left(x\right)$. Find the equation of $g\left(x\right)$ in the form $\log_2\left({ax+b}\right)$ where $a,b\in \mathbb{Z}$