Question

What are the phase shift and period for the function y = 3cos[4(θ + 45°)] − 4?

Phase shift = right 45°, period = 180°
Phase shift = right 45°, period = −180°
Phase shift = left 45°, period = 45°
Phase shift = left 45°, period = 90°

Answer 1

`\bf ~~~~~~~~~~~~\textit{function transformations} \\\\\\ % function transformations for trigonometric functions % templates f(x)={{ A}}sin({{ B}}x+{{ C}})+{{ D}} \\\\ f(x)={{ A}}cos({{ B}}x+{{ C}})+{{ D}}\\\\ f(x)={{ A}}tan({{ B}}x+{{ C}})+{{ D}} \\\\ -------------------\\\\`


`\bf \bullet \textit{ stretches or shrinks}\\ ~~~~~~\textit{horizontally by amplitude } |{{ A}}|\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }{{ B}}\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis}`


`\bf \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}\\\\ \bullet \textit{vertical shift by }{{ D}}\\ ~~~~~~if\ {{ D}}\textit{ is negative, downwards}\\\\ ~~~~~~if\ {{ D}}\textit{ is positive, upwards}`


`\bf \bullet \textit{function period or frequency}\\ ~~~~~~\frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ ~~~~~~\frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta)\\\\ -------------------------------\\\\`


`\bf \textit{now, with that template in mind, let's see} \\\\\\ y=3cos[4(\theta +45)]-4\implies y=\stackrel{A}{3}cos(\stackrel{B}{4}\theta \stackrel{C}{+180})\stackrel{D}{-4} \\\\\\ \stackrel{phase~shift}{horizontal~shift}\qquad \cfrac{C}{B}\implies \cfrac{180}{4}\implies 45 \\\\\\ period\qquad \cfrac{2\pi }{B}\implies \cfrac{2\pi }{4}\implies \cfrac{\pi }{2}\implies 90^o`