WEBVTT

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Okay. We are going over the different properties of quadratics based on the graph,

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based on the equation or a certain form of the equation. What is

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the domain of all polynomials? All real numbers. Again, you could write it like that

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or you

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could write from negative infinity to positive infinity. Okay. Does this quadratic

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have a

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minimum or a maximum? Minimum. Minimum. Okay. So, and what is that minimum value?

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Negative one.

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Yes, negative one. So go to the min, circle the min, or whatever. Min equals

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negative one.

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Minimums and maximums are y values or function values. It's just a number.

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Therefore, we know

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the range. What's the range? Negative one all the way to infinity, including

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negative one.

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We think this is the y intercept, but we don't know yet for sure. Maybe it's just

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very close to

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seven. Can we tell what the zeros are yet? Yeah. No. No. We could approximate it,

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but we definitely can't tell. It's probably going to be some irrational numbers. Do

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we know the

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leading coefficient? Go over. So if we have the graph, go to the vertex, you go over

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one,

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and however many you go up or down, that is the coefficient. So we go over one, up

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one, two.

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So the leading coefficient, our a value is two. That's not on the list, but let me

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write it over

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here. Okay. And now we can write the equation for this in what form? The vertex

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form. We have the

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We know the vertex and the leading coefficient, so use vertex form:

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f(x)=2(x−(−2))²−1,

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or 2(x+2)²−1.

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Axis of symmetry. Actually, we could definitely have done that before. What is the

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axis of

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symmetry? It's an equation x equals negative two, the x coordinate of the vertex.

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We can put it in standard form. How do we do that?

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Yeah. Multiply this out. So we got

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Two times (x+2)(x+2), minus one.

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Inside, x squared plus two x plus two x plus four; then multiply by two and subtract

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one.

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Two x squared plus eight x plus eight minus one.

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There is our standard form: 2x²+8x+7.

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There are other ways to figure out these properties. There is more than one method.

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Now we know the y-intercept.

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seven. This guy right there. Seven.

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Okay. Or you could write zero comma seven. Either one's fine.

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Factor form. How are we going to do that?

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Okay. Okay. Yeah. Let's find the zeros, which are when we set this equal to zero.

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Go zeros. We set Y equal to zero.

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Solve. You can either solve by factoring, but I don't know if this is going to

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factor.

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Okay. In fact, if we can't figure out what those are, it's not going to factor

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easily. So we got to use quadratic formula.

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64 minus 56.

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Is that rad eight? Yeah.

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And rad eight simplifies to two rad two.

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And we could divide them both by two.

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Divide: negative eight over four, plus or minus two root two over four. Negative

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two, plus or minus

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root two over two. Is that right?

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You could write it all as one big fraction or two separate numbers like I did.

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So those are our zeros.

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Negative two plus rad two over two.

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And negative two minus rad two over two.

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Therefore, we can write it in factor form. What's factor form going to be?

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X plus two.

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F of X equals the leading coefficient, which is what?

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Two times X minus first zero, which so it's going to be X minus this whole thing.

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Subtract the whole zero: x minus (−2+√2/2). The first factor is x+2−√2/2.

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I will write it once so we do not have to do that again. The factors are x+2−√2/2

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and x+2+√2/2, with the leading coefficient two.

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Yeah, there's factor form. Last thing is new is called intervals of increase.

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So when is the function increasing and when is it decreasing?

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This part of the graph, is that increasing or decreasing?

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Okay, how do we know?

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Okay, what about this half?

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What is going up?

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Okay, so a function is increasing or decreasing from left to right.

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Okay, so here from left to right, the Y values are going down. And from here, left

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to right, the Y values are going up. They're increasing.

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It has to do with as X is increasing, what is Y doing?

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Okay, and the number line, X increases left to right. So as X from here on, as X is

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increasing, Y is increasing.

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So the interval of increase is going to be from negative two to infinity.

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Okay, that function is increasing on that interval.

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Okay, we'll talk about Y at negative two later.

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And then from left to right, it's decreasing from negative infinity to negative two.

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Okay, why do I include negative two in both of them?

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Okay, in my saying it's increasing at negative two.

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We're not talking about whether a function is increasing, it doesn't increase or

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decrease at a point.

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Okay, it's only increasing on an interval.

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Okay, because everything on that interval, which is an interval of X values, the Y

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value is getting larger.

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Okay, or the actual definition of increasing is if you have any two numbers in the

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set,

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okay, let's call it A and B,

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a and b, with a less than b, then f(a) is less than f(b).

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Okay, so if you take any two numbers in this set, this is going to be true.

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Okay, including at negative two, and the only possibility for negative two would be

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the A.

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Okay, you take any other number, this is going to be true.

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Long story short: if that does not make total sense, remember the intervals of

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increase

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and decrease, we include the endpoints. Okay, okay, let's go to the next problem.

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Sorry, what form is this in? Factor form. So where does this tell us immediately?

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Zero. Zeros, and they are negative one and three.

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Okay, what else do we know automatically? It opens down because the leading

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coefficient is negative one.

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We need to find some more properties. What should we do next?

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The axis of symmetry, we can do that. What's that going to be?

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So negative one plus three divided by two, the average of the zeros.

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Okay, which is

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two over two, which is one.

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Okay, so the axis of symmetry is an equation. X equals one. Okay, and how do I find

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the vertex now that we have the axis symmetry?

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You can plug it in.

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Substitute one: one plus one, and one minus three.

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Negative two times negative two, which is four.

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The vertex is

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one four.

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We do need the

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leading coefficient. Actually, do we know the leading coefficient? Yeah, it's

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negative one. So we can write vertex form now, right?

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f(x)=−(x−1)²+4.

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We know the domain. What's the domain? All real numbers.

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And the range, we said it has a maximum. Did we say that already? Yes. The maximum

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was the max.

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That was four. Therefore, we know the range. Four is the maximum.

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We need to get standard form.

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Distribute the product, then apply the outside negative sign. Inside the parentheses

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are x²−3x+x−3.

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That gives positive two x,

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plus three.

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The y-intercept is three.

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And lastly, the interval, where is it increasing? When is it going left to right

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uphill?

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Negative infinity to one.

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And decreasing on the other half.