Question

Describe the relationships between the graphs of f and g. Think about amplitudes, periods, and shifts.
f(x) = cos4x
g(x) = -2 + cos4x
a. g(x) is 2 units down compared to f(x).
b. The period of g(x) is twice of that of f(x).
c. g(x) is a vertical shift of 2 units downward.

Answer 1

i am here to save the day!

Answer 2

ok...

Answer 3

lol just kidding i have no idea how to solve this @mitchal

Answer 4

@Preetha @Nnesha @kiamousekia @nevermind_justschool

Answer 5

b isn't true. f and g have the same period

Answer 6

\(\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& Upwards
\\ \quad \\
{\color{green}{ D}} < 0 & Downwards\end{array}
\\

\qquad \downarrow\qquad\qquad\quad\ \downarrow\\
% template start
f(x) = {\color{purple}{ A}} ( {\color{blue}{ B}}x + {\color{red}{ C}} ) + {\color{green}{ D}}\\
% template ends
\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\ the\ left
\\ \quad \\
\frac{{\color{red}{ C}}}{{\color{blue}{ B}}} < 0& to\ the\ right\end{array}
\end{array}\\
-----------------------------------\\
\bf f(x)=cos({\color{blue}{ 4}}x)\qquad \qquad g(x)=-2+cos({\color{blue}{ 4}}x)\iff g(x)=cos({\color{blue}{ 4}}x){\color{green}{ -2}}\)

Answer 7

y = a cos b(x - c)) + d
a = amplitude
b = (2π)/period
c = phase shift
d = vertical shift

Answer 8

Is it c? But then what's the difference between c and a?

Answer 9

Hmmm tricky question... my guess would be that it isn't c because they didnt specify what it was shifted down from. Presumably they mean the origin, but because they didn't say it is ambiguous whereas a specifies the magnitude of the shift and gives a reference from where the pellet occurs

Answer 10

And since both amplitude and period are equal (with no phase difference) then at all points g(x) will be the exact same graph as f(x) only shifted downward by 2