Thing is being recorded. All right.
Nobody showed up, so that's alright. Go ahead and get started. If if y'all show up, then we will go from there. Okay. So what I'm gonna do is actually go ahead and share my screen.
And we're gonna go over the quadratic formula. So there's a method that will work when solving quadratics every time. So please remember, when you're solving a quadratic, you're looking for the solutions.
So, if the quadratic touches the X axis two times, that would be two solutions.
Is the quadratic only touches one time that would be one solution and then is the quadratic touches never touches. Then that would be no solutions.
So the quadratic formula we can use that to go ahead and solve any for any quadratic. If it's in standard form.
So is going to be X equals negative B plus or minus the square root of be squared minus where AC,
all of that divided by two a,
don't forget to a part because lots of people forget to divide all that by two.
It. Okay. So, how do I solve the quadratic for using the project formula? You're gonna be sure the equation is in standard forms.
We need a X squared plus B X plus C identify your ABC you're gonna plugin a B and C into the formula. And then you're going to simplify so let's take a look at a couple of example, problems.
So, we've got an example one X squared, minus five X, minus fourteen equals zero. My a value is one. It is still there even though you don't see, it might be is negative five. My is negative fourteen.
So, now I'm gonna use that formula that we talked about X equals negative, be plus or minus the square root of B squared minus for AC, all divided by two.
So I'm gonna enter in my values. So, my V value here.
Is negative five and so that's gonna go right here right here. My, a value is one, so that's gonna go right here. Right here and my C values. Negative fourteen. It's gonna go right there.
Okay. Everything else is going to stay the same. We're gonna have X.
equals negative and then negative five is our B value plus or minus the square root B negative five squared minus four times a
times.
C.
All divided by two times. A okay, so please notice I'm using parentheses when I replacing the letters with my number of value, right?
It's important that you do that, because if you are putting into a calculator, anything like that negative five squared is not the same as parentheses negative five squared. Okay. So please be careful. Make sure you use. Alright.
So now we're going to simplify.
So I'm gonna have X equals negative negative five would be a positive five plus or minus the square root. Okay.
So negative five squared is twenty five, minus four times fourteen times one. So that's gonna give me a negative four times.
One times native fourteen is gonna be plus fifty, six, all divided by two times one, which is two. Okay. So we're gonna do twenty five plus fifty six.
We're gonna simplify underneath the radical that's gonna give us eighty one. Okay, so, from here we're going to take the square root of eighty one.
Luckily, the square root of eighty one is positive or negative nine. Alright.
Alright, so we've got X equals five plus, or minus nine divided by to so, from here we're going to do is do this in a kind of to solution.
So, we've got X equals five plus nine divided by two and X equals five minus, nine divided by two.
Because that's what that plus or minus symbol means. So, X equals five. Plus nine would be for taking over to one solution is gonna be seven.
Okay and then five minus sign X equals.
Negative for over to so we're gonna get negative too. So those are our two solutions. Okay. So how can we verify that? We did that correctly? A quick way to do that is go ahead and type.
We're gonna type our original equation into Desmos.
If I can find it.
And so it's gonna be X squared minus.
X minus fourteen, we can take a look, our solutions are negative to and seven so yet we did get this problem. Correct? And that is how you use the quadratic formula. Okay.
So let's take a look at another example here. Alright. So, for this one, the issue here is we are not in standard form.
Okay so remember our, we're going to be sure the equation isn't standard form that's step one so that we can identify a B and C. okay. So we need to rearrange this equation to put it in standard form first.
So what I'm gonna do is to X squared plus X equals five. I need this five to move to the other side. So I'm gonna subtract five from both sides. So, I'm gonna get two X squared plus X, minus five equals zero. Okay.
We needed to be equal to zero. That's going to be our standard form in order to identify our ABC. Okay. Alright.
So, our value is to be value is one and our C value is negative five.
And now again, we're gonna use that formula X equals negative B plus, or minus the square root of B squared minus for AC, all divided by two and we're gonna fill in our letters.
So, negative is one plus or minus the square root of is one minus, four times a times. See all divided by two times. Okay.
All right from there we're going to go ahead and simplifies via X equals negative one plus or minus the square root. One squared is one minus four times.
Two times negative five is gonna be positive to forty, all divided by four. Okay.
So, from here, we're gonna get simplify X equals negative one plus or minus the square root for forty one over four. Alright. So, what is the square root? Forty one?
Well, in this case, we don't have a perfect square. Okay so it's gonna it's gonna be a little harder to simplify that.
Number, alright, so for this one, let's go ahead and check and check my solutions to plus X equals five.
Two squared X is five. Yeah, our solutions are decimal so that that's gonna be correct so if we put a negative one, plus.
Squared of forty one divided by. Was it for? Do we get one of our solutions? Yes, we do. Okay.
It's the next one point. Three five one, right?
This is, we're not gonna be able to simplify that further because we can't simplify the forty one right here. Right?
So that's gonna be our two solutions native one plus square to forty one divided by four and negative one minus square two, forty one divided by four.
Okay, so that is the quadratic formula and how you use it there's a piece underneath the quadratic formula.
Sure, nobody's in. No. Okay so there's a piece underneath the quarter formula. So if you scroll back up here.
If you take a look underneath, it's B squared minus for AC. So, this is a nifty handy thing called the discriminate. Okay. And this is going to tell us how many solutions we have.
So, as I was saying before, we've got your two solutions, your one solution. And you're no real solutions, right? So, if you are calculating, the discriminate that piece, that's underneath the radical in the quadratic formula.
If you end up with a number of greater than zero, you're going to have two, real solutions. So, two X intercepts.
Did you get zero you're going to have one real solution,
which is one X intersect and then if you get less than zero,
so a negative number you're gonna have to imaginary solutions,
which is you,
they do not have X intercepts,
but you still have imaginary solutions?
Okay, alright, so let's take a look at some examples so we've got this right here. We're gonna do the same thing we need to identify a B and C. okay.
And we're gonna use that piece of the formula X equals negative be plus or minus the square root E squared minus for AC all divided by two a,
we're just gonna take a look at this B squared minus for AC.
Okay. Alright. So, B's tens. We're gonna put ten squared minus four times a, which is five times see, which is five. Okay. Alright.
So we're going to have ten square, which is one hundred minus four times, five times five, which is one hundred. We're going to get a result of zero. Alright.
So, what that means coming back up here to these notes is our discriminate is zero. That means we have one.
Solution okay, so this is going to touch the X axis one time. So if I were to say, like, how many solutions does this quadratic have? You can use the discriminate the B squared minus for AC to quickly.
Tell me how many solutions this one is going to have one real solution. So, let me show you. We've got five square plus ten expose five.
Five X squared plus ten X plus five. Okay. Check it out. We have one real solution at negative one. Alright. Alright. So that is what the description is for. It's really handy. So, let's try another example again.
We need to make sure this is a standard form. It is, it does equals zero.
My a, is one might be is negative six and my is ten so we're gonna use that formula E squared minus for AC the discriminate to find how many solutions we've got.
Alright, so you're gonna put in be, which is negative six squared, minus four times a, which is one time C, which is ten.
So six, they squared negative, six, hundred and six, thirty six minus four times, one times, ten, which is forty. We're gonna get a result of negative for.
Alright, so we got a negative number so our discriminate is less than zero. So that means we're gonna have a imaginary solutions and no X intercepts. Okay.
So you could say, no real solutions.
Okay, I'm gonna show you what that looks like X squared minus X plus ten.
X squared minus six X plus ten check it out. We do not have an intercept, so we do not have any real. So this here. Okay. And the last example X squared minus six X plus eight.
So re, use that discriminate formula be squared minus for a.
C, identify your a, which is one your bees negative six, and is eight so negative six squared minus four times a.
Times see. Okay so we're gonna get thirty two so this would be thirty six minus thirty two. We get a result of four. Okay.
So this is a positive number looking back at our notes. If we get a, something that's greater than zero that's gonna be two solutions two X intercepts. Okay this is gonna be two real solutions.
Let's confirm that with our graph X squared minus six X, eight.
X squared minus six X eight, and we do get to a real solutions across the exit access twice. Okay. That is how you use.
The discriminate is just that little tiny piece under the radical in our quadratic formula and how it's helpful for determining how many solutions we should get when we use the quadratic formula.
All right and that is it for the quadratic formula go back.
Oh, look, it's my.
Oh, right no. Yeah.
Record it's recording.
Of the student.
Is.
All right guys, I'm just gonna work a couple of these problems for characteristics, some quadratic functions. So this is the graph just wants to know what the Y intercept is. So, if you take a look, this is we're looking for, is where does it touch the Y axis.
Okay so we cross the Y axis. It's also the vertex through the vertex is that higher low point so we crossed the Y axis at negative two.
So, that is going to be, are Y intercept. Okay. What are the coordinates of the vertex okay so your Vertex remember, that's your higher low point.
In this case it has high point a maximum that occurs at X is to comma and then why is negative one? So that is our Vertex to comma negative one.
Oh, no. That's not right to have to do to really go to one. Perfect. Okay.
What are again, what are the coordinates at the vertex? So, we're looking for the highest or lowest point. This one has a high point maximum and that's happening at zero for the X value and two for the Y value. So we've got zero.
Yeah, okay. Whereas the, why intercept crosses the Y, axis at zero zero? What is the equation of the axis of symmetry?
Okay so remember, the axis of symmetry is the line that goes down the middle of your it touches your vertex. Okay. So, it's kinda like a butterfly. It's where, if you took that line, you fold the quadratic over onto itself. It would be.
Exactly symmetrical. Okay. So that is happening. Our vertex is the exile. Your vertex is at to. So, our access inventory is gonna be X equals to remember it's an equation.
It's not just a number. So, make sure that you put your X equals.
Okay, what is the maximum value of the function? So, maximum means you're looking for the highest point and that's gonna be your Vertex up here and then the highest point when we're talking about height, you're gonna be looking at the Y value.
Okay, because the Y, axis is how tall you are, and that should be five, six and it looks like eight.
So that is our maximum value, you can hear my cat back that what is the maximum value of the functions?
So same thing here, what is the top of our quadratic that's gonna be the, why values? So it's can be zero and we are good to go.
Okay, here's another access symmetry question. So we're going to go look for over text is the line that goes down the center. Okay so this one's occurring it negative two.
So, it's gonna be X equals negative two. That's going to be our line of symmetry.
X.
Okay, and that is it for that one.
Let's just do a couple of quick ones for the next one.
Okay, so transformations of quadratic function. So what's happening here is we've got our for text form. Okay. And this is same translated to units down. Okay, so remember if you've got.
Hey, X minus H squared plus K, this value your a value is making it stretching it. So it's in here or fatter. Okay.
That's what the a values doing.
Your age value is moving things left, or right?
I can change the color. Okay. And then your K value is gonna move things up or down. Right? So if we're translating this four units down, that means, we're gonna wanna change our K value.
So we're gonna wanna add or subtract to our to our original parent function, which is this one right here. Okay. Alright. So if you've got X squared and we want to move something down, we're going to subtract from our X squared.
Alright. And we want to move it four units. So our new equation is gonna be X squared minus four.
Okay, so I wanna show you that in depth most really quick so that you can kind of get a visual. Alright.
So, if you have your parent function, X squared, it's always gonna be the original function if you're dealing with function. Okay. And then X squared minus four take a look, we did move down for units.
So we went from zero to negative for so nothing else changed. It's not stretcher compress, not skinnier, fatter their wider it's not left moving left to. Right? We just translated it down for. Alright. Let's try another one.
This one wants us to translate eight units to the right. So, like, I was saying before the age is the value that transfers things to the, to the right. So it's gonna be inside the parentheses that's all.
We're changing here and to go to the right. We're going to put minus eight because here's why the formula has a minus and it already. Alright.
So, the original in order to go to the right, we would have had to put in a positive eight because when you think going to the right and moving to the right you're going in a positive direction. Okay. So same here.
X minus eight and I'll show you in Desmos why? Okay so we're gonna move X minus eight.
Squared notice we went to the right eight, right? If I had done the opposite, if I did X plus a cause, that's kinda what you think of when you think right it actually goes to the left. Alright. So that's because the formula has a minus sign.
So, it would've been minus negative eight to get that to be a plus. Alright. So be careful when you're dealing with left and right. It's kind of the opposite of what you think.
Alright, and let's do one more. This one's a reflection across the access. Okay. So I was saying that the a value is what is stretching or compressing right?
Well, it also is going to reflect are the other way. Okay. So, in order to do that, we have to make a negative. So we're just going to our a value right now is one.
We're just going to make it a negative one.
In order to get it to reflect across the access. Okay. And that is it for today guys I hope this was informative. Please contact me and reach out if you need any help whatsoever. Okay. Thank you.
