Graphing sinusoidal functions with transformations
About this video
The video graphs H(x) by rewriting it as H(x) = 4 sin(3(x - pi/3)) + 1 and building it from y = sin(x) in steps. Replacing x with 3x makes it cycle three times faster, x - pi/3 shifts it right by pi/3, the 4 multiplies the amplitude by 4, and the +1 shifts it up by 1.
You'll be able to
- factor the inside of a sine function to reveal the horizontal shift.
- describe how a, b, h and k change the graph of y = a sin(b(x - h)) + k.
- track an extremum and a midline point through each transformation.
Key words
- amplitude
- How far a sinusoid rises above or falls below its midline.
- midline
- The horizontal line halfway between a sinusoid's maximum and minimum.
- period
- The horizontal length of one full cycle of the graph.
- extremum
- A maximum or minimum point on a graph.
- phase shift
- A horizontal shift of a sinusoidal graph.
Transcript
- [Instructor] We're told to graph H of X is equal to all of this in the interactive widget. Notice that the movable point of the extremum, so this is that, can only define an extremum, so it's a minimum or maximum point. And the other movable point is a neighboring midline section.
So it can be immediately to the right or the left of that extremum point. And we can see if this is the midline, it intersects that midline. So what I'm going to do here is try to break this down. Start with a simple sign function and then think about how I would transform it to get this expression here.
And one of the things that will be useful is actually, let me rewrite this expression here. If I were to rewrite it as H of X, H of X is equal to four times sine of, I'm going to factor out a three, three times X minus pi over three, and then I have a plus one.
Now why did I do this? Well, this is making it clear. Actually, let me just do the various complications or the various transformations. I could start with Y equals sine of X. We'll do that in a second. And then what I could then do is do Y is equal to sine of three X.
Replacing an X with a three X is essentially going to compress it in the horizontal direction by a factor of three. It's going to cycle three times faster. So I could write that. Cycle three times faster is one way to think about it. Or it's going to have one third of the period.
Then if I replace this X with an X minus pi over three, so then if I go to Y is equal to sine of three times X minus pi over three, if you replace an X with an X minus pi over three, this is going to shift to the right, to the right by pi over three. And then, let's see, what could we do next?
We can multiply it by four. So then if we have Y is equal to four times sine of three times X minus pi over three, then what is this going to do? Well, this is just going to increase the amplitude by four. So this is going to be four times the amplitude, amplitude.
And then last but not least, I keep writing the same function or the same expression or the same equation, four times sine of three times X minus pi over three. And then I'm going to have the plus one. And so what this is going to do is shift up by one, shift up by one.
So let's do each of those steps. So let's start with Y equals sine of X. So let's see sine of X, we need to. Sine of X at zero radiance is zero, sine of zero is zero, sine of pi is zero, sine of pi over two is one. So it should look like this, and sine of three pi over two is negative one.
So this is sine of x. So I've done this first one here. Now if I want to do sine of three x, I'm going to cycle three times faster. So instead of this local maximum point happening at pi over two, it would happen at pi over six, which is exactly halfway between here and here.
And then this intersection, this midline intersection should now happen at pi over three. So let's see, if I do this, looks like I got what I wanted. This is now at pi over six. It's halfway. These markers are exactly a third of a pi, and this is right in between two of these.
So this is at pi over six and this midline intersection is at pi over three instead of pi. So this is sine of three X right over here. That's this one. Now, I now want to do sine of three times x minus pi over three. So I'm going to shift to the right by pi over three.
And pi over three is exactly one of these boxes in the horizontal direction. So I'm going to shift both of these. I'm going to shift this one by pi over three and then I'm going to shift that one by pi over three. So what I have graphed right over here is all of this business.
Now to put this four in front, I'm going to multiply the amplitude by four. So the midline intersection is, four times zero is still zero, but when it's one, if you multiply that by four, you now are going to get right over there. And then last but not least, what I have graphed now is this thing, but now we want to shift everything up by one to have this plus one at the end.
So I'll move this one up by one and I will move that one up by one. And we're done. This is the graph of H of X.
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