Question

What is the amplitude, period, phase shift, and midline of f(x) = −3 sin(4x − π) + 2?

Answer 1

Answer:

$Amplitude=3$

$Period=\frac{\pi}{2}$

$\text{Phase shift}=\frac{\pi}{4}$

$Midline=2$

Step-by-step explanation:

The general sine function is defined as

$g(x)=A\sin (Bx+C)+D$          .... (1)

where, |A| is amplitude, $\frac{2\pi}{B}$ is period, -C/B is phase shift and D is midline.

The given function is

$f(x)=-3\sin (4x-\pi)+2$              .... (2)

On comparing (1) and (2), we get

$A=-3$

$B=4$

$C=-\pi$

$D=2$

Using these values we can say that

$Amplitude=|A|=|-3|=3$

$Period=\frac{2\pi}{B}\Rightarrow \frac{2\pi}{4}=\frac{\pi}{2}$

$\text{Phase shift}=-\frac{C}{B}\Rightarrow -\frac{-\pi}{4}=\frac{\pi}{4}$

$Midline=D=2$

Therefore, $Amplitude=3$, $Period=\frac{\pi}{2}$, $\text{Phase shift}=\frac{\pi}{4}$ and $Midline=2$.

Answer 2

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ % function transformations for trigonometric functions \begin{array}{rllll} % left side templates f(x)=&{{ A}}sin({{ B}}x+{{ C}})+{{ D}} \\\\ f(x)=&{{ A}}cos({{ B}}x+{{ C}})+{{ D}}\\\\ f(x)=&{{ A}}tan({{ B}}x+{{ C}})+{{ D}} \end{array}$

$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks}\\ \quad \textit{horizontally by amplitude } |{{ A}}|\\\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{vertical shift by }{{ D}}\\ \end{array}$

$\bf \begin{array}{llll} \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}\\ \qquad \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ \qquad \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta) \end{array}$


now, with that template above in mind, let's take a peek of yours

$\bf \begin{array}{lcllcclll} f(x)=&-3&sin(&4x&-\pi)&+2\\ &\uparrow &&\uparrow &\uparrow &\uparrow\\ &A&&B&C&D \end{array}$

now, as far as the midline, well, the midline for sin(x) is just the x-axis, notice this one, has a D vertical shift, so the midline moved up that much