Question

General equation f(x) = a sin b(x - c) + d

How do I investigate the effect on f(x) of changes om the value of the constants a, b, c and d?

Answer 1

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

\qquad \downarrow\qquad\qquad\quad\ \downarrow\\
% template start
y = {\color{purple}{ A}} sin( {\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}
}\)

Answer 2

and also for trigonometric functions, their period is \(\bf \cfrac{original\ period}{{\color{blue}{ B}}}\quad for\ sin,cos\quad \cfrac{2\pi}{{\color{blue}{ B}}}\quad for\ tan,cot\quad \cfrac{\pi}{{\color{blue}{ B}}}\)

Answer 3

Thank you. :)

Answer 4

yw

Answer 5

If you have a program like geogebra, you can plot the function y = a*sin(b*x - c) + d. It knows that x is the variable and everything else is a constant. What it will do from there is create number sliders for a,b,c,d and let you fiddle with those sliders. That way you can see how changing each value corresponds to visual graph shifts/shape changes.

I think desmos does the same more or less

https://www.desmos.com/calculator