TEF - Shifting Sine and Cosine Lesson

Shifting Sine and Cosine

So, we've stretched and compressed sine and cosine, and now it's time to shift the graphs! Let's compare the graphs of equation image indicator $LaTeX: f\left(x\right)=\sin x\:and\:f\left(x\right)=\sin\left(x+\frac{\pi}{4}\right)$ $f\left(x\right)=\sin x\:and\:f\left(x\right)=\sin\left(x+\frac{\pi}{4}\right)$ .

A function of the form $LaTeX: f\left(x\right)=\sin\left(bx+c\right)\:or\:f\left(x\right)=\cos\left(bx+c\right),\:then\:the\:phase\:shift=-\frac{c}{b}$ $f\left(x\right)=\sin\left(bx+c\right)\:or\:f\left(x\right)=\cos\left(bx+c\right),\:then\:the\:phase\:shift=-\frac{c}{b}$ ,

Watch this video to practice graphing a bit:

So, now let's talk about shifting sine and cosine up and down. Below, is an animation of two functions: equation image indicator $LaTeX: f\left(x\right)=\cos x\:and\:f\left(x\right)=\cos\left(x\right)+D$ $f\left(x\right)=\cos x\:and\:f\left(x\right)=\cos\left(x\right)+D$

Watch this video to practice graphing a bit:

Remember for f(t)=A sin(Bt+C)+D or f(t)=A cos(Bt+C)+D
amplitude=|A|
period=2pi/|B|
midline: y=D
distance between critical points = Period/4

With a function of the form, equation image indicator $LaTeX: f\left(t\right)=\sin\left(t\right)+D\:or\:f\left(t\right)=\cos\left(t\right)+D$ $f\left(t\right)=\sin\left(t\right)+D\:or\:f\left(t\right)=\cos\left(t\right)+D$ , the midline is LaTeX: y=D $y=D$ . Let's put it all together and practice what you've learned.

Application Problem:

The current, I , in amperes flowing through a particular alternating current circuit at a time, t seconds is: equation image indicator $LaTeX: I=-240\sin\left(22\pi t\right)$ $I=-240\sin\left(22\pi t\right)$ .

1. What is the amplitude of the current?

Solution: $LaTeX: amplitude\:=\:\left|-240\right|\:=\:240$ $amplitude\:=\:\left|-240\right|\:=\:240$

2. What is the period of the current?

Solution: $LaTeX: period\:=\frac{\:2\pi}{22\pi}=\frac{1}{11}\:\sec.$ $period\:=\frac{\:2\pi}{22\pi}=\frac{1}{11}\:\sec.$

IMAGES CREATED BY GAVS