Graphing Quadratic Functions
October 13, 2026

[00:00:05] I got two different equations up here. What do we call the one on the left? A linear equation.
[00:00:13] A linear equation. Why is it called a linear equation? Because the leading term
[00:00:23] has degree one. We have two different polynomials here, one on each side.
[00:00:35] All equations have an expression on the left and an expression on the right. Both of these are polynomials.
[00:00:44] This is what type of polynomial? Linear. It happens to be a binomial. This is a what? Constant. The degree is one, the degree is zero. So the larger degree determines what we call the entire equation.
[00:01:08] So this is called linear equation. We could call it a linear polynomial equation, but we just linear implies it's a polynomial. Why is this called a quadratic equation?
[00:01:32] The polynomial on the left is what type of polynomial? Quadratic. The polynomial on the right is what type? Linear. The larger of the two is the quadratic. So we call the whole thing a quadratic equation.
[00:01:55] Are the solutions to these two equations, are they numbers or variables? Numbers. Why are they just numbers? There's only one variable.
[00:02:10] So this is a linear equation in one variable. This is a quadratic equation in one variable. The solutions are numbers. And how would we solve this one? Isolate the variable. We can't do that here. But how would we solve this? We've got three options.
[00:02:32] Quadratic formula, factoring, and completing the square. Now I've got two more equations. Are the solutions to these two numbers? Are they numbers?
[00:02:58] No, why not?
[00:03:01] So the solutions are ordered pairs.
[00:03:08] What do we call this type of equation?
[00:03:22] This is what type of polynomial? Linear. What is this type of polynomial? Linear.
[00:03:35] But because y equals some other expression of x, we call this a linear function.
[00:03:54] And this is the same exact equation except using different notation. Y and f of x mean the same thing. Why do we like this notation better? Or at least why do I like this notation? You might not like it.
[00:04:12] Or is there a reason?
[00:04:17] Let me plug in x equals 3 here. Now let's go with 5. We get y equals 3 times 5 minus 2. Y equals 13.
[00:04:38] And let me plug in 5 into this equation.
[00:04:51] We get f of 5 equals 13. Why is this better than this?
[00:05:02] Does this show you what polynomial?
[00:05:05] Sure. Does this show you what we plug in to get 5?
[00:05:09] Well, not what variable, but what value of x goes with this y value?
[00:05:18] So we know this equation definitely has the solution 5, 13. It's not obvious what x value we plugged in unless you look back here and know that that was where we plugged in.
[00:05:34] So the function notation tells you what we plugged in for x to get the y value or the f of x value. It basically tells you the ordered pair. That's why we like that notation better.
[00:05:47] Does that make sense?
[00:05:49] Typically, once we got y equals, we would call this a quadratic function, which tells us the two variable equation.
[00:06:04] And this is the same equation, same function as the same solution. I'm just using different notation.
[00:06:12] How many solutions does this guy have?
[00:06:25] Uh-oh. One or two? I heard infinitely many.
[00:06:33] Oh boy. We just found one solution. What was it? 5, 13?
[00:06:40] How could I find another one? Plug in any x value. Let's go with zero.
[00:06:50] We get f of zero is negative 2, which tells us zero, negative 2 is the solution.
[00:06:58] So this linear equation has how many solutions?
[00:07:02] Infinitely many. You can plug in any value for x and get a y value.
[00:07:09] And so we don't list all the solutions. What do we do with all the solutions?
[00:07:22] Or how do I write the solutions? And we graph them.
[00:07:31] And that linear function, that linear equation, that graph are all of its, or is the solution set.
[00:07:44] Every ordered pair on that line, red line, is a solution. Every ordered pair not on that line is not a solution.
[00:07:50] This make sense?
[00:07:53] Okay. So what are we going to do with this? How are we going to write the solution set here?
[00:08:02] Graph it.
[00:08:03] We're going to graph the solution set.
[00:08:07] Uh-oh. How are we going to graph this?
[00:08:14] Y intercept.
[00:08:19] We do know where the y intercept, where's the y intercept?
[00:08:21] Negative 4.
[00:08:22] Negative 4. Plug in zero.
[00:08:24] Zero.
[00:08:26] Now what?
[00:08:27] For all of this. Plug in the number for x. We get like two points.
[00:08:34] We can plug another number in. I like this. What's another number?
[00:08:37] One.
[00:08:45] What is that?
[00:08:50] Negative 7?
[00:08:51] I was negative 8.
[00:08:53] Negative 8? I mean?
[00:08:55] Yeah.
[00:08:55] Yes.
[00:08:57] Okay.
[00:08:57] Try negative 1?
[00:08:58] Yeah.
[00:08:59] Sure.
[00:09:07] What is that? 1 plus 5?
[00:09:13] 6 minus 4 is 2. Is that right?
[00:09:16] We need to do this, please.
[00:09:18] Okay, now we're there. Uh-oh. Do they fall on the line?
[00:09:22] No, it is going to curve.
[00:09:24] They definitely don't fall on the line.
[00:09:27] There's a reason we call this one a linear equation.
[00:09:31] Because all the ordered pairs fall on the line.
[00:09:34] It's definitely not a linear equation.
[00:09:41] Uh, it's some curve that's not a line.
[00:09:45] We do have a specific name for that curve. We'll tell you later.
[00:09:48] Before we graph this one, let's graph the mother of all quadratic functions.
[00:09:56] Why did I call it the mother of all quadratic functions?
[00:10:01] It's actually known as the parent function.
[00:10:04] Okay.
[00:10:06] Uh, once we get the graph of this, all quadratics are going to look very, very similar.
[00:10:14] They're just shifted and stretched a little bit.
[00:10:17] Okay.
[00:10:18] So this is the, called the parent function of all quadratic functions.
[00:10:22] And let's just, uh, just do a bunch of ordered pairs.
[00:10:31] x and f of x. Or x and y. Or f and x and x squared.
[00:10:43] Or the y coordinate, same thing.
[00:10:46] Uh, let's start with zero.
[00:10:49] Zero.
[00:10:49] What is zero squared?
[00:10:51] Zero.
[00:10:53] Zero.
[00:10:53] Uh, let me do this over here.
[00:11:05] Uh, let me call this f of x.
[00:11:10] Zero squared zero.
[00:11:12] Uh, what is one squared?
[00:11:15] Two squared?
[00:11:17] Four.
[00:11:17] Four.
[00:11:18] Three squared?
[00:11:19] Nine.
[00:11:21] Nine.
[00:11:23] Four squared?
[00:11:24] Sixteen.
[00:11:25] Sixteen.
[00:11:26] Okay.
[00:11:28] And let's plug in some negative values.
[00:11:30] Let's go negative one.
[00:11:32] One.
[00:11:33] One.
[00:11:34] Negative two?
[00:11:39] Negative three?
[00:11:43] I don't know why I'm putting them in parentheses.
[00:11:49] Square.
[00:11:54] So these are just a few of the infinite solutions.
[00:11:58] Let's go ahead and plot them.
[00:12:02] One squared is one.
[00:12:04] Two squared is four.
[00:12:07] Three squared is nine.
[00:12:26] Four squared is sixteen.
[00:12:28] That's off our graph.
[00:12:31] And it's a very smooth curve.
[00:12:46] Ah, I screwed up.
[00:13:03] It's almost like a U looking thing.
[00:13:09] So that red are all the solutions to that quadratic function y equals x squared or f of x equals x squared.
[00:13:20] All quadratic are going to look something like this, a U.
[00:13:24] It's just going to be stretched or flipped or translated left or right, up and down and so on.
[00:13:34] So this specific shape is called a parabola.
[00:13:39] The shape is called...
[00:13:46] Okay.
[00:13:48] It's called a parabola.
[00:13:49] Only quadratics form parabolas. Just because another function looks like a U does not make it a parabola.
[00:13:58] It's not a parabola.
[00:14:04] What other properties, just by looking at this, does this have...
[00:14:10] Does this have a minimum value or a maximum value?
[00:14:15] This minimum, or if it's flipped over, you'll see later it'll have a maximum.
[00:14:21] This point's called the vertex.
[00:14:25] Okay.
[00:14:26] It's the same name as what we call these things in a triangle.
[00:14:31] Vertex, vertices, plural.
[00:14:36] When we graph absolute value functions, what do we call this?
[00:14:41] The vertex.
[00:14:42] Okay.
[00:14:43] Parabolas have a vertex as well.
[00:14:48] This case is 0, 0 is the vertex.
[00:14:56] Is this a symmetric function?
[00:15:00] Yes.
[00:15:00] It is.
[00:15:04] Symmetric functions have what we call an axis of symmetry.
[00:15:21] I already spelled it.
[00:15:25] Okay.
[00:15:26] This one's axis of symmetry, so it's a vertical line.
[00:15:31] How would we describe the vertical line?
[00:15:36] You guys did...
[00:15:37] When you did linear functions later on, this vertical line is not a function.
[00:15:42] It's a line.
[00:15:43] What was the equation for this one right here?
[00:15:47] For this one in red is x equals negative 6.
[00:15:51] Okay.
[00:15:52] All vertical lines is x equals a number.
[00:15:56] It's kind of like 0, y plus x equals negative 6.
[00:16:05] So what is the axis of symmetry for this quadratic?
[00:16:10] Not zero.
[00:16:11] x equals zero.
[00:16:13] It's an equation.
[00:16:14] It's not a number.
[00:16:16] Okay.
[00:16:17] That vertical line is x equals zero.
[00:16:23] The third...
[00:16:24] Well, we'll talk about it.
[00:16:25] Don't let me forget the domain range.
[00:16:30] These have what we call symmetric points.
[00:16:46] Okay.
[00:16:47] And what are symmetric points?
[00:16:48] There are two points that are equal distance from the axis of symmetry.
[00:16:56] So this is of distance two away.
[00:16:59] This is the same distance two away from the axis of symmetry.
[00:17:04] So any functions that are symmetric have symmetrical points,
[00:17:09] and they're symmetric about the axis of symmetry.
[00:17:14] Those are the main properties.
[00:17:15] What is the domain of this function?
[00:17:25] Domain is the set of all x coordinates.
[00:17:30] Are there any x values we can't plug in here?
[00:17:33] No, we can plug in any real number.
[00:17:36] Okay.
[00:17:37] So the domain is all real numbers or the set of real numbers.
[00:17:46] But the range here is not all real numbers.
[00:17:52] Okay.
[00:17:53] This has a minimum.
[00:17:55] What is the minimum value?
[00:18:00] Minimums and maximums are numbers.
[00:18:03] Zero is the minimum.
[00:18:05] So when we do our interval notation of the range,
[00:18:08] it has a minimum.
[00:18:11] And is zero in the range?
[00:18:13] Yes.
[00:18:14] It is.
[00:18:15] Okay.
[00:18:16] Is there a maximum?
[00:18:17] No, it goes up forever.
[00:18:19] So zero to infinity, including zero.
[00:18:22] It's the range.
[00:18:32] Okay.
[00:18:34] That's all the properties.
[00:18:37] me do minus four.
[00:18:53] Is this a quadratic function?
[00:18:54] Yes.
[00:18:57] Y equals or F of X equals some quadratic.
[00:19:06] How are we going to graph this?
[00:19:13] We could start by plugging in a bunch of values.
[00:19:17] Okay.
[00:19:22] Let me go backwards.
[00:19:25] I'll come back to this.
[00:19:26] You don't need reds.
[00:19:36] We graphed lots of absolute-value functions a while ago.
[00:19:42] Remember this?
[00:19:42] Six.
[00:19:44] Did we just plug in x values to find y?
[00:19:47] No.
[00:19:48] No.
[00:19:48] How did we go about graphing it?
[00:19:53] The plus three is like minus.
[00:19:56] You just got to do the thing.
[00:19:57] Do the thing?
[00:19:59] What do we call the thing?
[00:20:03] We looked at the different transformations, right?
[00:20:09] When you replace x with x plus three, where is that shifted?
[00:20:15] To the left by three.
[00:20:18] So it shifts absolute value of X left by three.
[00:20:23] And if you add or subtract a number outside of the function, what does that do?
[00:20:30] Lowers down.
[00:20:31] Lowers down.
[00:20:32] So this goes down by five.
[00:20:40] And was there any stretch to it?
[00:20:42] No.
[00:20:43] No reflection or stretch because the coefficient from the function is a one.
[00:20:48] Okay.
[00:20:49] So we found the vertex.
[00:20:51] Left three.
[00:20:53] Down five.
[00:20:55] The vertex.
[00:20:56] And now what do we do?
[00:21:00] Draw.
[00:21:01] Draw what?
[00:21:02] Two lines.
[00:21:03] Yeah, this was a piecewise function.
[00:21:07] This half has a slope of what?
[00:21:11] This half has a slope of negative one.
[00:21:14] Negative one.
[00:21:16] What if this were two?
[00:21:21] What would we do?
[00:21:23] Well, let's move on.
[00:21:24] Two is positive two.
[00:21:26] Yeah, this stretches this function by a factor of two.
[00:21:32] Okay.
[00:21:32] Because they're linear, we just change the slopes.
[00:21:35] We're going to do something different with quadratics.
[00:21:39] We're going to do the same thing.
[00:21:40] We're going to use transformations.
[00:21:42] Okay.
[00:21:44] The first thing we're going to do is find the vertex.
[00:21:51] Well, actually, we'll talk about that in a second.
[00:21:54] For x squared, right?
[00:21:56] The vertex is a zero zero.
[00:21:59] Okay.
[00:22:01] What is this minus four going to do?
[00:22:03] Four.
[00:22:04] Five.
[00:22:07] All this is going to do is take the graph of x squared.
[00:22:11] It goes down by four.
[00:22:14] So the vertex is going to be (0, negative 4).
[00:22:22] The vertex.
[00:22:28] Okay.
[00:22:31] What I like to do is instead of graphing this.
[00:22:48] We're going to follow a pattern.
[00:22:52] I'll show you in a second.
[00:22:56] And this is my pattern.
[00:22:59] I'm calling the pattern.
[00:23:03] Zero squared is zero.
[00:23:05] One squared is one.
[00:23:06] Two squared is four.
[00:23:08] Three squared is nine.
[00:23:15] We could subtract four from all the y's.
[00:23:24] Zero.
[00:23:25] Zero minus four is negative four.
[00:23:29] One minus four.
[00:23:31] Four minus four.
[00:23:34] Oops.
[00:23:35] Yeah. Nine minus four.
[00:23:38] The negative four.
[00:23:39] Negative three.
[00:23:40] Zero.
[00:23:45] Five.
[00:23:48] But with graph paper, we do not need to bother with all of this.
[00:23:54] I just call this.
[00:24:02] This is my vertex.
[00:24:07] Okay.
[00:24:08] And here's what I do.
[00:24:11] I start at the vertex.
[00:24:14] This is my starting point for the zero-zero offsets.
[00:24:17] And I go.
[00:24:19] One to the right.
[00:24:20] One up.
[00:24:22] So we go one to the right, one up.
[00:24:24] This is going to be my next point.
[00:24:26] I go back to the vertex.
[00:24:27] Go two to the right, four up.
[00:24:30] One, two, one, two, three, four.
[00:24:35] Three to the right from the vertex, nine up.
[00:24:38] One, two, three.
[00:24:39] One, two, three, four, five, six, seven, eight, nine.
[00:24:46] That's what I call my pattern.
[00:24:49] Go to our vertex, which is the zero zero.
[00:24:51] Go one to the right, one up, two to the right, four up, three to the right, nine up, and so on.
[00:24:56] And then this is the right half of the parabola.
[00:25:01] What's the axis of symmetry here?
[00:25:05] X equals zero.
[00:25:05] X equals zero because we did not shift it left and right.
[00:25:13] So I usually put a dotted line for my axis of symmetry.
[00:25:20] And then we use the fact that it's a line.
[00:25:25] It has symmetric points.
[00:25:28] So any X value, we go same.
[00:25:31] So I went one here and I go one over.
[00:25:35] Two, two, three, one, two, three.
[00:25:40] And there we got enough points.
[00:25:44] So once we find the vertex, we follow the pattern.
[00:25:48] One to the right, one up, two to the right, four up, three to the right, nine up.
[00:25:52] And then we just graph these symmetric points.
[00:25:55] Once we have to add, draw a nice smooth curve.
[00:26:09] And there is the graph of the parabola.
[00:26:12] That is the solution to it.
[00:26:15] It should be quite good.
[00:26:16] Make sense?
[00:26:18] Okay.
[00:26:21] Let us look at another one.
[00:26:22] Is this a quadratic function?
[00:26:25] Yes.
[00:26:27] This is some linear binomial square gives us a quadratic.
[00:26:32] And then we subtract three.
[00:26:34] This is not written in standard form.
[00:26:37] What's standard form?
[00:26:43] The form that we use for the quadratic formula.
[00:26:46] A x squared plus b x plus c.
[00:26:49] Standard form.
[00:26:50] We'll talk about this form in a minute.
[00:26:54] Where's the shift or function X squared?
[00:27:00] Because I replaced X with X minus two.
[00:27:04] What's the shift?
[00:27:06] Left or right?
[00:27:08] Right.
[00:27:08] Right by two.
[00:27:12] And this shifts it down by three.
[00:27:21] So this vertex goes to the right, two, down three.
[00:27:29] So the vertex is.
[00:27:34] Two, negative three.
[00:27:38] Okay.
[00:27:41] Once I find the vertex, I follow my pattern.
[00:27:46] And the pattern is zero, zero, one, one, two, four, three,
[00:27:53] square is nine.
[00:27:56] So I go to the vertex.
[00:27:58] I go to the right one, up one, right two,
[00:28:04] up four, right three, up nine, one, two, three, four, five,
[00:28:11] six, seven, eight, nine.
[00:28:16] And then what do we do?
[00:28:25] We graph these symmetric points along the axis of symmetry.
[00:28:31] The axis of symmetry always goes through the vertex.
[00:28:34] So I got a symmetric point here.
[00:28:38] One, two, one, two.
[00:28:41] One, two, three, one, two, three.
[00:28:45] And now I just draw a nice, smooth curve.
[00:28:52] And the very bottom at the vertex, it becomes very, very flat.
[00:28:57] It is not a sharp V. It is very smooth.
[00:29:01] So don't draw Vs.
[00:29:05] There it is.
[00:29:14] Real quick, what's its domain?
[00:29:22] Yeah, all real numbers.
[00:29:25] Like all polynomials, polynomial functions, the domain is all real numbers.
[00:29:31] Does this have a minimum or a maximum?
[00:29:34] It says a minimum and the minimum is?
[00:29:40] Negative three.
[00:29:42] It is always the y coordinate of the vertex.
[00:29:47] So therefore the range has a minimum.
[00:29:53] Does not have a maximum.
[00:30:04] This is not a standard form.
[00:30:07] Anybody want to guess what form it's in?
[00:30:10] It is called vertex form.
[00:30:14] This is called vertex form of a quadratic function.
[00:30:26] If it's in this form, it's very easy graphically.
[00:30:29] Instantly know the vertex.
[00:30:31] It is the opposite of this number, comma that number.
[00:30:38] We usually call the vertex h,k.
[00:30:46] I do not know why those letters were chosen.
[00:30:49] There they are.
[00:30:51] So vertex form is y equals a.
[00:31:03] We'll talk about it in a second.
[00:31:04] X minus h, squared, plus k.
[00:31:15] The x coordinate of the vertex is the opposite of this number, which is just h.
[00:31:23] The y coordinate is just that, whatever that number is.
[00:31:33] Is this the same?
[00:31:36] So standard form.
[00:31:41] Is y equals a x squared plus b x plus c.
[00:31:50] Is this a in vertex form is same as that a in standard form?
[00:31:55] It is.
[00:31:57] Once you multiply out this binomial squared, you get x squared plus something plus something.
[00:32:04] And then you distribute the a.
[00:32:07] It's the same as in standard form.
[00:32:11] It is the leading coefficient.
[00:32:16] So, let's try another one.
[00:32:18] What form is this in?
[00:32:21] Vertex form.
[00:32:23] Okay.
[00:32:25] It takes our x squared, just to the left by three, up by one.
[00:32:31] So our vertex is a negative three one.
[00:32:33] Once I get my vertex.
[00:32:35] Once I get that, what do I do?
[00:32:44] Follow the pattern.
[00:32:47] What do I call it?
[00:32:48] Zero, zero, one squared is one, two squared is four, three squared is nine.
[00:32:53] So go to the vertex.
[00:32:57] One to the right, one up.
[00:33:01] Two to the right, one, two, three, four up.
[00:33:05] Three to the right, nine up.
[00:33:13] That's going to be the right half of our parabola.
[00:33:17] And how do I find the left half?
[00:33:22] Use the what points?
[00:33:25] Symmetric points.
[00:33:32] And draw a nice smooth curve.
[00:33:34] Very good.
[00:33:35] Yes.
[00:33:36] This is a vertex.
[00:33:37] Yes.
[00:33:40] Therefore the vertex is.
[00:33:46] Opposite that number, comma, that number, negative one, negative five.
[00:33:54] So go ahead on your graph paper.
[00:33:58] Draw a couple axes.
[00:34:25] That's what I call my pattern.
[00:34:29] It's just zero, one, two and three squared.
[00:34:37] This leading coefficient of two.
[00:34:40] What is that going to do to it?
[00:34:43] It's going to stretch it by a factor of two.
[00:34:48] So.
[00:34:51] Here's what we do.
[00:34:59] We simply multiply these Y values by two.
[00:35:04] So times two is.
[00:35:07] Let me just do a whole nother one.
[00:35:09] X and two x squared.
[00:35:14] So zero zero.
[00:35:18] Multiplying by two gives the offsets (0,0), (1,2), (2,8), and (3,18).
[00:35:30] Now this is my stretched pattern.
[00:35:35] This is my vertex right here.
[00:35:39] And now I go one to the right, two up.
[00:35:43] One to the right, two up.
[00:35:48] Two to the right, eight up.
[00:35:57] Three to the right, eighteen.
[00:35:59] I don't know if we're going to get there.
[00:36:11] Oh.
[00:36:12] Oh.
[00:36:14] It's up here somewhere.
[00:36:28] Now what?
[00:36:31] We have our right half of our parabola.
[00:36:35] We have the symmetric points.
[00:36:38] We plot the symmetric points.
[00:36:41] About this line.
[00:37:18] And there it is.
[00:37:19] It's just a skinnier looking parabola.
[00:37:24] It's been stretched.
[00:37:30] Okay.
[00:37:31] Questions on this so far?
[00:37:34] Let's try it on.
[00:37:52] The vertex is.
[00:37:56] Negative one.
[00:37:58] The opposite of that number.
[00:37:59] Comma three.
[00:38:03] This is the vertex.
[00:38:23] Okay.
[00:38:25] So the one half is going to compress it or stretch it by a factor one half.
[00:38:33] What does the negative do?
[00:38:40] Negative inside of a function.
[00:38:42] Or sorry, outside of a function.
[00:38:44] It reflects it.
[00:38:47] But.
[00:38:48] We take care of that very easily.
[00:38:53] X and negative one half X squared.
[00:38:57] You just multiply these all by.
[00:39:00] Negative one half.
[00:39:13] Let me add one more for 16.
[00:39:20] Negative one half.
[00:39:26] Literally we are just multiplying this guy by negative one half to get this.
[00:39:32] We get zero zero one negative one half.
[00:39:37] Two negative two three negative nine halves.
[00:39:44] Which is negative four point five.
[00:39:52] OK, so this is our new pattern.
[00:39:59] We multiply the leading coefficient by all the y values in the pattern.
[00:40:04] So this is our vertex, the starting point for the zero-zero offsets.
[00:40:07] Go one to the right.
[00:40:10] But now we go since it's negative.
[00:40:12] We go down by a half.
[00:40:16] Two to the right down by two.
[00:40:19] Three to the right down by four and a half.
[00:40:23] We go one to the right down a half.
[00:40:29] Two to the right down to.
[00:40:35] Three to the right.
[00:40:37] Down four and a half.
[00:40:39] One, two, three, four.
[00:40:42] And a half.
[00:40:44] And four to the right, down eight.
[00:40:46] One, two, three, four.
[00:40:57] Now I have my right hand side.
[00:41:06] This is the axis of symmetry.
[00:41:11] And a nice curve—do not do that yet.
[00:41:15] I need my symmetric points.
[00:41:49] This.
[00:41:51] Not too bad.
[00:41:54] OK, we find the vertex.
[00:41:57] We take our pattern zero zero one one two four three nine.
[00:42:01] Sometimes add four sixteen.
[00:42:03] Then we multiply these y values by the leading coefficient.
[00:42:08] This gives us our new pattern.
[00:42:10] When it's negative, we just go down.
[00:42:12] We always go to the right.
[00:42:15] And now we just go down by whatever this is.
[00:42:18] If it's negative values.
[00:42:32] Now let's do something fun.
[00:42:35] This is not a vertex form.
[00:42:37] What form is this in? Standard form.
[00:42:47] Actually, let me change this.
[00:43:16] Actually, let's leave it.
[00:43:17] Sorry.
[00:43:19] We factor it out.
[00:43:23] Let's try it.
[00:43:25] What is the factor to.
[00:43:33] OK, it could be factored, but we have to use the quadratic formula.
[00:43:39] OK, we're going to get two irrational.
[00:43:44] Numbers right here.
[00:43:47] Actually, let's let's pretend it actually factors it.
[00:43:53] It doesn't.
[00:43:54] But let's say it did.
[00:44:04] What would the numbers three and negative two be?
[00:44:12] OK, let's.
[00:44:16] If I plug in three into here.
[00:44:19] I get three minus three times three plus two, which is equal to zero times five, which is zero.
[00:44:32] OK, so we know three comma zero is a solution.
[00:44:38] If I plug in negative two.
[00:44:43] Be negative two minus three.
[00:44:47] Times negative two plus two.
[00:44:52] Negative five times zero, which is zero.
[00:45:02] This is not the function, but let's say it was.
[00:45:13] So that parabola would now look something like this.
[00:45:18] What do we call the points that intersect the X axis?
[00:45:25] X intercepts gives us the X intercepts.
[00:45:27] So if we're able to factor it, it only gives the X intercepts.
[00:45:33] It does give us two symmetric points.
[00:45:40] So if these were the irrational factors, it gives the X intercepts.
[00:45:45] In fact, that has a specific name.
[00:45:47] We'll get to that later.
[00:45:50] So.
[00:45:53] Just factoring is not going to help.
[00:46:03] What do I know the leading coefficient is?
[00:46:11] What's the leading coefficient?
[00:46:14] One.
[00:46:15] OK.
[00:46:24] If I could get it into this form, what do we call this form?
[00:46:28] Vertex.
[00:46:29] Vertex form.
[00:46:31] We would know where the vertex was and could easily graph it.
[00:46:41] I need to make a perfect square.
[00:46:45] Binomial.
[00:46:48] From this.
[00:46:50] How am I going to make this a perfect-square binomial?
[00:46:58] By completing the square.
[00:47:06] So here's what we're going to do.
[00:47:07] So write this.
[00:47:10] X squared minus 2X.
[00:47:12] Leave some space.
[00:47:15] And put a minus 5.
[00:47:16] Why? I'm not going to bring the 5 over here.
[00:47:22] Why am I not going to do that?
[00:47:25] We don't want 5 plus f of X.
[00:47:27] We just want f of X equals.
[00:47:29] So we just leave it to the right here.
[00:47:34] Oops.
[00:47:35] Minus 5.
[00:48:02] So what number am I going to add here?
[00:48:07] We take this number.
[00:48:10] When we complete the square,
[00:48:12] Divide by 2.
[00:48:16] And then do what with it?
[00:48:18] Square it.
[00:48:22] Okay.
[00:48:25] And I'm going to add 1.
[00:48:28] And we're not going to add 1 to both sides.
[00:48:32] We could.
[00:48:33] You can't just add 1.
[00:48:35] But you can add 1.
[00:48:37] And then subtract 1.
[00:48:40] Just adding 0.
[00:48:43] Now this is a perfect square.
[00:48:47] And it's going to factor to what?
[00:48:49] X minus 1 times X minus 1.
[00:48:53] And now the negative 1 minus 5 is negative 6.
[00:49:07] Okay. Now it's in vertex form.
[00:49:13] Okay.
[00:49:19] Now we know the vertex.
[00:49:24] What is the vertex?
[00:49:28] 1 negative 6.
[00:49:34] Once I have the vertex.
[00:49:37] Who do I do?
[00:49:41] We follow the pattern.
[00:49:47] 0 0 1 1 2 squared is 4 3 squared is 9.
[00:50:14] I'm not following the pattern.
[00:50:20] 9 1 2 3 4 5 6 7 8 1.
[00:50:28] I'll get my symmetrical points.
[00:50:36] Okay.
[00:50:50] Okay.
[00:50:58] If you do not like completing the square, there is another option.
[00:51:04] Okay.
[00:51:24] So where did we get this number from?
[00:51:26] The minus 1.
[00:51:34] We took this number.
[00:51:37] Divided by 2.
[00:51:39] We squared it to figure out what we had, right?
[00:51:43] But simply that number divided by 2.
[00:51:47] Okay.
[00:51:55] And then of course, once we get it into here,
[00:51:59] we know the vertex is the opposite of that number.
[00:52:03] Okay.
[00:52:04] So it turns out the vertex, the x coordinate,
[00:52:13] is negative b over 2.
[00:52:19] Okay.
[00:52:20] Unless there's a leading coefficient,
[00:52:23] then we'd have to divide everything by the leading coefficient.
[00:52:27] So it turns out it's negative b over 2a.
[00:52:31] I could show you another reason for that.
[00:52:38] And then how do I get the y coordinate?
[00:52:46] If we have standard form,
[00:52:52] this is the x coordinate.
[00:52:55] We simply plug it into the function.
[00:53:01] Okay.
[00:53:02] So let me show you how to do that.
[00:53:06] Let's do another example.
[00:53:08] What form is this in?
[00:53:11] Standard form.
[00:53:11] Actually, let me change this y to f of x.
[00:53:17] Standard form directly shows two things.
[00:53:22] It tells us the y intercept.
[00:53:25] The only other thing it tells us,
[00:53:27] does it open up or does it open down?
[00:53:31] If the leading coefficient is positive,
[00:53:33] it's going to open up.
[00:53:35] Leading coefficient is negative.
[00:53:36] Okay.
[00:53:38] So the vertex is always at negative b over 2a,
[00:53:48] f of negative b over 2a.
[00:53:55] So negative b over 2a is negative.
[00:54:01] What's our b value?
[00:54:03] Negative 6.
[00:54:04] The negative of negative six, all over two times a.
[00:54:06] What's our a?
[00:54:08] 1.
[00:54:09] Which is 6 over 2, which is 3.
[00:54:16] So I now know my vertex is at 3, f of 3.
[00:54:24] And how do I find my f of 3?
[00:54:27] You just plug it into our function.
[00:54:31] F of 3 is 3 squared minus 6 times 3 plus 5.
[00:54:51] Is that negative 4?
[00:54:57] So my vertex is at 3, negative 4.
[00:55:03] Once we know that, we know our function in vertex form is the leading coefficient.
[00:55:13] What's the leading coefficient?
[00:55:15] 1.
[00:55:16] 1.
[00:55:17] So it's x minus 3 squared minus 4.
[00:55:23] It's the opposite of that number.
[00:55:29] And now we can graph it quickly.
[00:55:37] We go to 3.
[00:55:44] Negative 4.
[00:55:47] Since the leading coefficient is positive, we follow the pattern.
[00:56:19] I'll write my symmetric points.
[00:56:34] So if it's in standard form,
[00:56:42] just write it in vertex form by finding the vertex.
[00:56:46] The vertex’s x coordinate is always negative b over two a.
[00:56:49] Once you get that number, we plug it back into our standard form to get the y coordinate.
[00:56:56] And then we write it in vertex form, or you just go with the vertex, follow the pattern.
[00:57:02] If we have to adjust the pattern because a is not 1, then we do that.
[00:57:07] Sound good?
[00:57:08] Yes, it is.
[00:57:09] Okay, last question before we start homework.
[00:57:14] Okay, here, I don't give you an equation.
[00:57:18] I give you a graph of a parabola.
[00:57:21] How are we going to find the equation?
[00:57:27] What is the vertex?
[00:57:28] Negative 2, 3.
[00:57:33] Negative 2, 3.
[00:57:36] Okay, vertex form is a x minus h squared plus k.
[00:57:50] So we know y equals a times x plus two, squared, plus three.
[00:57:57] And then how do I figure out my leading coefficient?
[00:58:03] We know it's negative.
[00:58:09] Okay, you see one point.
[00:58:12] We go to the right one and we go down one.
[00:58:17] That tells you the leading coefficient.
[00:58:19] We went down one, leading coefficient negative one.
[00:58:22] If we had to go down two, it would be negative two.
[00:58:27] Okay, it's all based on going over one.
[00:58:31] Do we go up one?
[00:58:32] Do we go up two?
[00:58:32] Do we go down one, down two, whatever it is.
[00:58:34] So we know it is y equals negative one times x plus two, squared, plus three.
[00:58:43] If we want standard form, we just have to multiply this out.