Quadratic Properties in Three Forms
October 19, 2026

[00:00:05] Okay. We are going over the different properties of quadratics based on the graph,
[00:00:14] based on the equation or a certain form of the equation. What is
[00:00:22] the domain of all polynomials? All real numbers. Again, you could write it like that or you
[00:00:30] could write from negative infinity to positive infinity. Okay. Does this quadratic have a
[00:00:41] minimum or a maximum? Minimum. Minimum. Okay. So, and what is that minimum value? Negative one.
[00:00:50] Yes, negative one. So go to the min, circle the min, or whatever. Min equals negative one.
[00:01:03] Minimums and maximums are y values or function values. It's just a number. Therefore, we know
[00:01:13] the range. What's the range? Negative one all the way to infinity, including negative one.
[00:01:22] We think this is the y intercept, but we don't know yet for sure. Maybe it's just very close to
[00:01:28] seven. Can we tell what the zeros are yet? Yeah. No. No. We could approximate it,
[00:01:38] but we definitely can't tell. It's probably going to be some irrational numbers. Do we know the
[00:02:00] leading coefficient? Go over. So if we have the graph, go to the vertex, you go over one,
[00:02:09] and however many you go up or down, that is the coefficient. So we go over one, up one, two.
[00:02:15] So the leading coefficient, our a value is two. That's not on the list, but let me write it over
[00:02:22] here. Okay. And now we can write the equation for this in what form? The vertex form. We have the
[00:02:32] We know the vertex and the leading coefficient, so use vertex form: f(x)=2(x−(−2))²−1,
[00:02:44] or 2(x+2)²−1.
[00:02:51] Axis of symmetry. Actually, we could definitely have done that before. What is the axis of
[00:02:55] symmetry? It's an equation x equals negative two, the x coordinate of the vertex.
[00:03:04] We can put it in standard form. How do we do that?
[00:03:09] Yeah. Multiply this out. So we got
[00:03:16] Two times (x+2)(x+2), minus one.
[00:03:24] Inside, x squared plus two x plus two x plus four; then multiply by two and subtract one.
[00:03:34] Two x squared plus eight x plus eight minus one.
[00:03:50] There is our standard form: 2x²+8x+7.
[00:03:52] There are other ways to figure out these properties. There is more than one method. Now we know the y-intercept.
[00:04:07] seven. This guy right there. Seven.
[00:04:13] Okay. Or you could write zero comma seven. Either one's fine.
[00:04:19] Factor form. How are we going to do that?
[00:04:26] Okay. Okay. Yeah. Let's find the zeros, which are when we set this equal to zero.
[00:04:36] Go zeros. We set Y equal to zero.
[00:04:45] Solve. You can either solve by factoring, but I don't know if this is going to factor.
[00:04:50] Okay. In fact, if we can't figure out what those are, it's not going to factor easily. So we got to use quadratic formula.
[00:05:03] 64 minus 56.
[00:05:08] Is that rad eight? Yeah.
[00:05:13] And rad eight simplifies to two rad two.
[00:05:26] And we could divide them both by two.
[00:05:36] Divide: negative eight over four, plus or minus two root two over four. Negative two, plus or minus
[00:05:50] root two over two. Is that right?
[00:05:54] You could write it all as one big fraction or two separate numbers like I did.
[00:06:01] So those are our zeros.
[00:06:10] Negative two plus rad two over two.
[00:06:16] And negative two minus rad two over two.
[00:06:20] Therefore, we can write it in factor form. What's factor form going to be?
[00:06:30] X plus two.
[00:06:37] F of X equals the leading coefficient, which is what?
[00:06:43] Two times X minus first zero, which so it's going to be X minus this whole thing.
[00:06:56] Subtract the whole zero: x minus (−2+√2/2). The first factor is x+2−√2/2.
[00:07:09] I will write it once so we do not have to do that again. The factors are x+2−√2/2 and x+2+√2/2, with the leading coefficient two.
[00:07:21] Yeah, there's factor form. Last thing is new is called intervals of increase.
[00:07:31] So when is the function increasing and when is it decreasing?
[00:07:44] This part of the graph, is that increasing or decreasing?
[00:07:50] Okay, how do we know?
[00:07:53] Okay, what about this half?
[00:07:58] What is going up?
[00:08:02] Okay, so a function is increasing or decreasing from left to right.
[00:08:12] Okay, so here from left to right, the Y values are going down. And from here, left to right, the Y values are going up. They're increasing.
[00:08:27] It has to do with as X is increasing, what is Y doing?
[00:08:34] Okay, and the number line, X increases left to right. So as X from here on, as X is increasing, Y is increasing.
[00:08:49] So the interval of increase is going to be from negative two to infinity.
[00:09:07] Okay, that function is increasing on that interval.
[00:09:13] Okay, we'll talk about Y at negative two later.
[00:09:17] And then from left to right, it's decreasing from negative infinity to negative two.
[00:09:29] Okay, why do I include negative two in both of them?
[00:09:40] Okay, in my saying it's increasing at negative two.
[00:09:44] We're not talking about whether a function is increasing, it doesn't increase or decrease at a point.
[00:09:52] Okay, it's only increasing on an interval.
[00:09:58] Okay, because everything on that interval, which is an interval of X values, the Y value is getting larger.
[00:10:05] Okay, or the actual definition of increasing is if you have any two numbers in the set,
[00:10:15] okay, let's call it A and B,
[00:10:19] a and b, with a less than b, then f(a) is less than f(b).
[00:10:29] Okay, so if you take any two numbers in this set, this is going to be true.
[00:10:37] Okay, including at negative two, and the only possibility for negative two would be the A.
[00:10:43] Okay, you take any other number, this is going to be true.
[00:10:47] Long story short: if that does not make total sense, remember the intervals of increase
[00:10:54] and decrease, we include the endpoints. Okay, okay, let's go to the next problem.
[00:11:06] Sorry, what form is this in? Factor form. So where does this tell us immediately?
[00:11:13] Zero. Zeros, and they are negative one and three.
[00:11:23] Okay, what else do we know automatically? It opens down because the leading coefficient is negative one.
[00:11:34] We need to find some more properties. What should we do next?
[00:11:39] The axis of symmetry, we can do that. What's that going to be?
[00:11:45] So negative one plus three divided by two, the average of the zeros.
[00:11:51] Okay, which is
[00:11:57] two over two, which is one.
[00:12:03] Okay, so the axis of symmetry is an equation. X equals one. Okay, and how do I find the vertex now that we have the axis symmetry?
[00:12:16] You can plug it in.
[00:12:21] Substitute one: one plus one, and one minus three.
[00:12:25] Negative two times negative two, which is four.
[00:12:38] The vertex is
[00:12:46] one four.
[00:12:47] We do need the
[00:12:55] leading coefficient. Actually, do we know the leading coefficient? Yeah, it's negative one. So we can write vertex form now, right?
[00:13:07] f(x)=−(x−1)²+4.
[00:13:16] We know the domain. What's the domain? All real numbers.
[00:13:22] And the range, we said it has a maximum. Did we say that already? Yes. The maximum was the max.
[00:13:31] That was four. Therefore, we know the range. Four is the maximum.
[00:13:41] We need to get standard form.
[00:13:47] Distribute the product, then apply the outside negative sign. Inside the parentheses are x²−3x+x−3.
[00:14:08] That gives positive two x,
[00:14:12] plus three.
[00:14:14] The y-intercept is three.
[00:14:27] And lastly, the interval, where is it increasing? When is it going left to right uphill?
[00:14:38] Negative infinity to one.
[00:14:49] And decreasing on the other half.

[00:14:51] Board review (no narration): x²−4x+6=(x−2)²+2. Vertex (2,2); range [2,infinity); no real zeros. Decreasing to x=2, then increasing.