00:01
So in this problem, we're given three different sinusoidal equations here with different shifts in it.
00:09
Two of them are on sine, one of them is on cosine.
00:12
And we're asked for the amplitude, the axis of curve, the range, the period, the phase shift, and the zeros in one period.
00:29
So let's start with this explanation first of all, for we look at any of those.
00:35
I think of this as y equals a sine bx plus c plus d like this.
00:47
Okay.
00:48
Or i could think of this as y equals a cosine bx plus c plus d.
00:58
Well, the first thing is that a tells me the amplitude, where the base function has a equal to 1.
01:20
Because when i graph a sign or a cosine, it goes from 1 to minus 1 back and forth.
01:26
And so its amplitude is always 1.
01:29
Okay.
01:31
Then b is working with the period, or the period is 2 pi over b.
01:46
So if i'm given b, then the period, is 2 pi over b divided by whatever that value b is.
02:04
Okay.
02:06
Now then, c, well, c is the phase shift.
02:22
If, let's look at our equations here, well, only one of them has something.
02:30
If c is negative, we shifted to the right.
02:44
If it's positive, well then we shifted left.
02:56
Okay.
02:59
And then d is the vertical shift.
03:08
Positive shifts negative shifts us down.
03:23
Okay, so with that, the first thing we're asked is the amplitude.
03:28
Okay, so we said the amplitude was a, whatever the value is for a, because the base function is a 1.
03:34
All right, so that means my amplitude here then, for this first one, is four from that.
03:43
Amplitude here is six, and amplitude here is 25.
03:48
We don't do the negatives because that just flips the function over is all that does.
03:57
Okay.
03:59
So then let's look at the range.
04:04
The range is the output, right? range is the output.
04:17
Okay.
04:18
Well, if i have an amplitude of four on this first one, and notice that there is a vertical shift here, right, shift up or down, of five, and i went down five.
04:38
So if i think about this for a minute on the graph, so i shifted down five, right? so i'm down here at negative five.
04:50
And my amplitude is 4.
04:54
So that means i'm going to go up 4 and down 4.
04:58
So that means my range is going to go from minus 1 to minus 9, isn't it? i have an amplitude of 4, which means i go up 4 or down 4 from where i'm the center, which is, by the way, the axis of this curve, which means this is y equals negative 5, by the way.
05:24
Okay.
05:28
Now then, let's look at the next one.
05:33
So again, we have an amplitude of 6, which means we're going to go up 6 or down 6, and we shifted down 3.
05:40
So negative 3 plus 6, that's the top of 3, isn't it? and then negative 3 minus 6 would be minus 9, wouldn't it? okay.
05:53
And this means then also that our axis of the curve is negative 3.
06:01
Okay, same thing here.
06:02
I'm going to go down 21 by that, by that shift, by that vertical shift there.
06:15
And i have an amplitude of 25.
06:18
So minus 21 plus 25 gives me a top of 4, doesn't it? and then the minimum down to bottom will be minus 21 minus 25, so that's minus 46.
06:33
So that means my axis of curve then is it y equals negative 21 because i did a vertical shift down.
06:39
Of 21.
06:42
Okay, remember the period now.
06:44
The period, right, is 2 pi over b...