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Okay. If you use the square root principle, you get y is plus or

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minus the square root of negative x squared plus nine, which is this and this. These

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two are functions.

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This is not a function. Is this a quadratic function?

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Is the square root of x a polynomial? That same thing is x to what power? One half

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power. Is that a polynomial?

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The exponents of the variables have to be whole numbers for it to be a polynomial.

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These square-root functions are not polynomials.

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So therefore these are functions, but they are not polynomial functions.

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So this is not a function. We could break it up into two separate functions, but it

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is a relation.

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Okay. And it has lots of solutions. Y squared negative x squared plus nine. Let's

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and we often actually write like this.

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X squared plus y squared equals nine. We'll come back to that. What is this gonna

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look like? Let's take a look.

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It turns out x squared plus y squared equals nine is a circle. And what's the radius

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of the circle?

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Three. So why in the world does this give us a circle? I'll kind of give you an

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idea. If I take the square root of both sides.

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Does this thing look familiar from last year or the year before?

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What if I ask you to find the distance between zero, zero, and x, y? What's the

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distance formula?

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If we have two ordered pairs, this is the change in x. This is the change in y.

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Okay. So the distance between them is basically the Pythagorean Theorem.

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The difference of the x's, the difference of the y's squared plus squared is the

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Pythagorean Theorem, which gives us this. Anyways, long story short, this is the

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distance formula between these two ordered pairs, zero, zero, and the ordered pair

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x, y.

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So everything is a solution to this. If it's the distance from zero, zero, the

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distance is equal to three. In fact, that is what a circle is. The set of ordered

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pairs is the same distance away. In this case, zero, zero.

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Anyways, long story short, this is a circle with radius the square root of nine, or

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in other words, three, centered at zero, zero.

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Let me give you a formula for a circle in general. It is (x minus h) squared plus (y

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minus k) squared equals r squared.

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So this is a circle with radius r centered at h, k.

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So, write the equation.

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So, this is a circle centered at three, negative two with a radius of five.

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Where does the three go? Five squared, which is twenty-five. That's it.

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Okay, really, it's just a, this is the transformation, the x direction, so to the

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right by three, up by two, or up by, sorry, up by negative two, which means down by

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two.

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Let's try another one.

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What is the center and radius?

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So, this is x minus what squared? Zero.

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And so, the center is zero, negative five, and the radius is the square root of

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seven.

48
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Okay, not bad. It's always the opposite of this number.

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Can I write down this formula? Yeah. It's x minus the x coordinate, y minus the y

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coordinate.

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So, it's always just the opposite of that number. So, the opposite is negative five.

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Okay, find the center and radius.

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So, we need to get it in this form. x plus or minus a number, y plus or minus a

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number, equals some number.

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How are we going to do that?

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First, let's rearrange it.

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So, I just grouped the x's together, the y's together, and brought the constant to

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the other side.

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I need to take this and factor it to a perfect square binomial. What do we call that

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process?

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Completing the square. So, we just got to add the correct number to make the same

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binomial, and so we can write it like that.

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So, how do I complete this square here?

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We take the negative 4 and divide it. Divide by 2 and square.

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And we add that to both sides.

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And then we do the same thing here. Square. 10. Divide by 2. Square.

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That's 25.

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So, the center is the opposite of that number.

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Square root of 26. Just barely larger and higher.

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Okay, let's try another one.

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Okay.

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So, what do we do here? We rearrange everything.

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This is called general form. We want to get into standard form.

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So, we're going to complete the square. So, what do we add?

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6 divided by 2. 9.

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Negative 12 divided by 2.

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So, we're going to do the same thing here.

78
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Negative 3, 6.

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And the radius.

80
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Okay, what questions do we have on this?

81
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We just have to complete the square twice to figure it out. It's not too

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complicated.