Question

rewrite y=sqrt(9x+45) -2 to make it easy to graph using a translation. Describe the graph

Answer 1

\(\large\color{black}{y=\sqrt{9x+45}-2 }\)
\(\large\color{black}{y=\sqrt{9(x+5)}-2 }\)
\(\large\color{black}{y=3\sqrt{x+5}-2 }\)

Answer 2

You simply factorize the k in 9x+45 so you can see the exact transformation of c.

Answer 3

Thanks a lot sir

Answer 4

So you can see that you are adding 5 "inside the parenthesis of the x". (in this case inside the square root)
and you can see the 3 in front of the square root of x, which is a stretching coefficient, which is greater than (or the absolute value of which is greater, I should say) than 1.
and finally, you are subtracting 2 from the entire input.

Answer 5

so it is shifted 5 units left and 2 units down on the graph?

Answer 6

Just to give you a couple of shift examples.

\(\large\color{ darkviolet }{\large {\bbox[5pt, yellow ,border:2px solid white ]{ \large\text{ }\\
\begin{array}{|c|c|c|c|}
\hline \texttt{Shifts} ~~~\tt from~~~ {f(x)~~~\tt to~~~g(x)}&~\tt{c~~~units~~~~} \\
\hline
\\f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x \normalsize\color{red }{ -~\rm{c}} } &~\rm{to~~the~~right~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x \normalsize\color{red}{ +~\rm{c}} } &~\rm{to~~the~~left ~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x} \normalsize\color{red}{ +~\rm{c} } &~\rm{up~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x} \normalsize\color{red}{ -~\rm{c} } &~\rm{down~} \\ \\
\hline
\end{array} }}}\)

\(\large\color{ darkteal }{\large {\bbox[5pt, cyan ,border:2px solid white ]{ \large\text{ }\\
\begin{array}{|c|c|c|c|}
\hline \texttt{Shifts} ~~~\tt from~~~ {f(x)~~~\tt to~~~g(x)}&~\tt{c~~~units~~~~} \\
\hline
\\f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= (x \normalsize\color{red}{ -~\rm{c} })^2 &~\rm{to~~the~~right~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= (x \normalsize\color{red}{ +~\rm{c} })^2&~\rm{to~~the~~left ~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= x^2 \normalsize\color{red}{ +~\rm{c} } &~\rm{up~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= x^2 \normalsize\color{red}{ -~\rm{c} } &~\rm{down~} \\ \\
\hline
\end{array} }}}\)

Answer 7

yes 5 left and 2 down.

Answer 8

And lastly, the stretch.

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

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

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 \(\normalsize\color{black}{ \rm{f(x)} }\). )


OR,

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

where the same conditions ONE and TWO are met.

Answer 9

if c is negative, then it is just another reflection across the x-axis and then the stretch.