This is a short week, but we should cover the end of 4.5 and the beginning of 4.6 before (or during) our Thanksgiving.
In section 4.5 we learn about the graph of sines and cosines. There a few new vocabulary items to describe the graphs. First though graph the function $y=sin(x)$ with your calculator and notice the following
In fact every $2\pi$.
the domain is all values of $x$
The range is only $[-1,1]$.
The fact that it repeats makes it ideal to describe things like the seasons, or the way a wave comes in.
There is some new notation to describe the curve $y = d + a\sin(bx + c)$
This is half the distance between the peak and the valley. It is $|a|$ in the above. So $y=5\sin(x)$ has an amplitude of 5, $y = -10\sin(x)$ has an amplitude of 10.
This is how long it takes to repeat. For the sine curve it is $2\pi$. However, if you change the $b$ from 1 this changes things. The new period is
$$T = \frac{2\pi}{b}$$
So if you graph $y = sin(2x)$ you will see it repeats every $2\pi/2 = \pi$ units, and the graph $y = \sin(\pi x)$ repeats every $2\pi/\pi = 2$ units.
The graph can be shifted up and down the $y$ axis with the value of $d$. This is the same thing as what happens when we compare the graph of $f(x)$ to that of $f(x) + c$.
This moves the graph left or right. If we have the graph $y=sin(x+5)$ then it is a sine graph shifted left 5 units. If we have the graph $y = sin(\pi x + \pi)$ then this is trickier. The period is $2\pi/b = 2$ the phase shift is left because of the plus sign, but the amount is not $c=\pi$ but rather $c/b = \pi/\pi = 1$. This is because the function $f(x)$ to which we investigate $f(x+c)$ is not $\sin(x)$, but rather $\sin(\pi x)$.
The goal of this section is to be able to look at the function $y = d + a\sin(bx+c)$ and tell what the graph should be without graphing, or to take a graph and figure out the values of $a,b,c$ and $d$.
So let’s just practice this way. With you calculator verify this
| $y = 5\sin(x)$ | sine curve with amplitude 5 |
| $y = 5 + \sin(x)$ | sine curve shifted up 5 units |
| $y = \sin(5x)$ | sine curve with period $2\pi/5 = 1.257..$ |
| $y = \sin(x+5$ | sine curve shifted left by 5. Note (0,0) moves to (-5,0) |
| $y = 3\sin(4x)$ | sine curve with period $2\pi/4 = \pi/2$ with amplitude 3. |
In this section you learn how to graph the other trig functions. The most important being the tangent function.
First, plot $\tan(x)$ on your calculator. You will notice:
It repeats itself. It does this every $\pi$ units. (not $2\pi$ like the sine).
At $\pi/2 = 90^\circ$ the tangent function is 1/0 which is undefined. As a consequence you get vertical asymptotes. The domain is all x except those of the form $k\pi + \pi/2$ for integer values of $k$.
At these asymptotes the function is unbounded. It can be any large or small number. Its range is $(-\infty,\infty)$.
Just like the sine function (and the cosine) you can stretch and shift the graph easily by multiplying and adding terms. In the homework, make sure you figure out where things go.