Question

In what direction and by how many units is the graph of f(x) = −7 sin(3x + π) − 2 vertically and horizontally shifted?

Answer 1

$\bf ~\hspace{10em}\textit{function transformations} \\\\\\ \begin{array}{llll} f(x)= A( Bx+ C)^2+ D \\\\ f(x)= A\sqrt{ Bx+ C}+ D \\\\ f(x)= A(\mathbb{R})^{ Bx+ C}+ D \end{array}\qquad \qquad \begin{array}{llll} f(x)=\cfrac{1}{A(Bx+C)}+D \\\\\\ f(x)= A sin\left( B x+ C \right)+ D \end{array} \\\\[-0.35em] ~\dotfill\\\\ \bullet \textit{ stretches or shrinks horizontally by } A\cdot B\\\\ \bullet \textit{ flips it upside-down if } A\textit{ is negative}$

$\bf ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if } B\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis} \\\\ \bullet \textit{ horizontal shift by }\frac{ C}{ B}\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is positive, to the left}$

$\bf \bullet \textit{ vertical shift by } D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{ B}$

now, with that template in mind

$\bf f(x) = \stackrel{A}{-7}sin(\stackrel{B}{3}x+\stackrel{C}{\pi } )\stackrel{D}{-2}\qquad \begin{cases} D=&-2\\ &\textit{vertical shift}\\ &\textit{downwards by 2 units}\\ \frac{C}{B}=&\frac{\pi }{3}\\ &\textit{horizontal shift}\\ &\frac{\pi }{3}\textit{ units to the left} \end{cases}$