Respuesta :
[tex]\bf \qquad \qquad \qquad \qquad \textit{function transformations}
\\ \quad \\
% function transformations for trigonometric functions
\begin{array}{rllll}
% left side 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}}
\end{array}\qquad[/tex]
[tex]\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks}\\ \quad \textit{horizontally by amplitude } |{{ A}}|\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \end{array}[/tex]
[tex]\bf \begin{array}{llll} \bullet \textit{vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}\\ \qquad \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ \qquad \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta) \end{array}[/tex]
now, with that template in mind, let's see
[tex]\bf \begin{array}{llccll} cos(&4x&+0)&+2\\ &\uparrow &\uparrow &\uparrow \\ &B&C&D \end{array} \\\\\\ period\qquad \cfrac{2\pi }{B}\implies \cfrac{2\pi }{4}\implies \cfrac{\pi }{2} \\\\\\ \textit{horizontal/phase shift}\qquad \cfrac{C}{B}\implies \cfrac{0}{4}\implies 0\impliedby none \\\\\\ \textit{vertical shift}\qquad D=+2\impliedby \textit{2 units up}[/tex]
now the range, how far up and down it goes on the y-axis
well, for the graph of cos(x), the range is, goes up to 1, down to -1, is all,
the midline is at 0
now, with a vertical shift of 2 upwards, it moves the midline by 2 units, used to be at 0, now is at y = 2, but the amplitude never changed, is goes up and down still one unit,
but with the midline at 2, goes up to 3 and down to 1, so the range is 3 ⩽ y ⩽ 1
[tex]\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks}\\ \quad \textit{horizontally by amplitude } |{{ A}}|\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \end{array}[/tex]
[tex]\bf \begin{array}{llll} \bullet \textit{vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}\\ \qquad \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ \qquad \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta) \end{array}[/tex]
now, with that template in mind, let's see
[tex]\bf \begin{array}{llccll} cos(&4x&+0)&+2\\ &\uparrow &\uparrow &\uparrow \\ &B&C&D \end{array} \\\\\\ period\qquad \cfrac{2\pi }{B}\implies \cfrac{2\pi }{4}\implies \cfrac{\pi }{2} \\\\\\ \textit{horizontal/phase shift}\qquad \cfrac{C}{B}\implies \cfrac{0}{4}\implies 0\impliedby none \\\\\\ \textit{vertical shift}\qquad D=+2\impliedby \textit{2 units up}[/tex]
now the range, how far up and down it goes on the y-axis
well, for the graph of cos(x), the range is, goes up to 1, down to -1, is all,
the midline is at 0
now, with a vertical shift of 2 upwards, it moves the midline by 2 units, used to be at 0, now is at y = 2, but the amplitude never changed, is goes up and down still one unit,
but with the midline at 2, goes up to 3 and down to 1, so the range is 3 ⩽ y ⩽ 1
