Question

Find the equation of a sine wave that is obtained by shifting the graph of y=\sin(x) to the right 2 units and downward 9 units and is vertically stretched by a factor of 6 when compared to y=\sin(x).

Answer 1

y = A sin( 2pi / period ( x - p.s.) ) + v.s.

Answer 2

\(\large {
\textit{function transformations}
\\ \quad \\

\begin{array}{llll}

\begin{array}{llll} shrink\ or\\ expand\\ by\ {\color{purple}{ A}}\cdot {\color{blue}{ B}}\end{array}
\qquad
\begin{array}{llll} vertical\\ shift\\ by \ {\color{green}{ D}} \end{array}

\begin{array}{llll}{\color{green}{ D}} > 0& Up\\ {\color{green}{ D}} < 0 & Down\end{array}
\\

\qquad \downarrow\qquad\qquad\quad\ \downarrow\\
y = {\color{purple}{ A}} ( {\color{blue}{ B}}x + {\color{red}{ C}} ) + {\color{green}{ D}}\\
\qquad\qquad\quad\ \uparrow \\

\qquad\begin{array}{llll} horizontal\\ shift\\ by \ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\end{array}

\begin{array}{llll}\frac{{\color{red}{ C}}}{{\color{blue}{ B}}} > 0 & to \ left\\ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}} < 0& to \ right\end{array}
\end{array}
}\)

any ideas?

Answer 3

the transformations are the same for say Asin(Bx+C) + D
thus

Answer 4

@jdoe0001 is the period 2?

Answer 5

period for sin(x) and cos(x) and is \(\large 2\pi\) if that's what you mean

Answer 6

none of the transformations affected the period of the sine curve

Answer 7

@jdoe0001 I'm confused what to plug in

Answer 8

yeah. you're not asked for a period
just a horizontal shfit of 2
a vertical shift of 9
and a stretch of 6

Answer 9

what you're asked is
what would the value for "C" be
and the value of "A"
and the value of "D"

Answer 10

@jdoe0001 so is B 6?

Answer 11

well. B would be 1
so \(\large \bf y = {\color{purple}{ A?}} sin( {\color{blue}{ 1}}x + {\color{red}{ C?}} ) + {\color{green}{ D?}}\)

Answer 12

notice, is going 2 units to the right, so C is a negative, or C < 0
is also going downwards 9 units, so D is negative as well, or D < 0
and has a stretch factor of 6, so A should be obvious

Answer 13

@jdoe0001 since 2 is going to the right wouldn't that mean positive?

Answer 14

one would expect that, yes
but the graph behaviour is that, if it's negative, it goes to the right

Answer 15

@jdoe0001 I GOT IT THANK YOU SO SO MUCH!

Answer 16

yw