Question

Functions f(x) and g(x) are shown below:

f(x) g(x)
f(x) = -4(x - 6)2 + 3 g(x) = 2 cos(2x - π) + 4


Using complete sentences, explain how to find the maximum value for each function and determine which function has the largest maximum y-value

Answer 1

have you covered function transformations yet?

should be in your book

Answer 2

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

\begin{array}{llll}

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

\begin{array}{llll}{\color{green}{ D}} > 0& Upwards\\ {\color{green}{ D}} < 0 & Downwards\end{array}
\\

% template start
\qquad \downarrow\qquad\qquad\quad\ \downarrow\\
y = {\color{purple}{ -4}} ( x - {\color{red}{ 6}} )^2 + {\color{green}{ 3}}\\
%template end
\qquad\qquad \quad \uparrow \\

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

\begin{array}{llll}{\color{red}{ C}} > 0 & to\ the\ left\\ {\color{red}{ C}} < 0& to\ the\ right\end{array}
\end{array}
\\ \quad \\
--------------------------------------------
\\ \quad \\


\begin{array}{llll}

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

\begin{array}{llll}{\color{green}{ D}} > 0& Upwards\\ {\color{green}{ D}} < 0 & Downwards\end{array}
\\

% template start
\qquad \downarrow\qquad\qquad\quad\ \downarrow\\
y = {\color{purple}{ 2}}cos ( {\color{blue}{ 2}} x - {\color{red}{ \pi }} ) + {\color{green}{ 4}}\\
%template end
\qquad\qquad \quad \uparrow \\

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

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

Answer 3

for the maximum y-value, the only transformation that matters are the vertical shift
and in the g(x) case, the "amplitude", or A component

Answer 4

well, for g(x) the "A"mplitude and the vertical shift, because the amplitude is a shift as well for trigonometric ones

Answer 5

Try using the second derivative test