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Ladies and gentlemen. I got a lot of questions off of this. And whatss even more disturbing.

Though is i got a lot of non questions that people got h.s or even tease. Ladies and gentlemen.

If you have a question. Please write that question on your homework list because otherwise. I dont know what else i what to answer the question.

I dont know where your misunderstanding is or not all right. So. If you have a misunderstanding.

Please make sure you guys write that down or if you have a question. I want to do the prom. So the question asks they give us a function f of x.

Equals. Negative. X.

Squared. Plus. 6x minus 15.

And what they ask is to do for this problem is to find the maximum in the minimum point. And then also determine the domain in the range all right so if you didnt write this down. Alex youre probably gonna want to write this down a sheet of paper.

So you have it either in your notes or not all right. So. The first thing we look at this ladys ellen is remember we got to talk about what do quadratics produce.

We wrote down in our notes that quadratics produce a graph that we call a parabola. Thank you so our graph is either going to open up or its going to open down. Its going to look something like that and then what we talked about was the maximum point was right there or it has a minimum point.

Its either one or the other so when you need to do is determine what is that maximum point all right so to do that if you guys remember that point is what we call the what shonn. You remember maximum point. Which we call the started with av vertex right so how did we find the vertex well what we remembered was the vertex went through the axis of symmetry.

So the first thing. I want to do is determine what is the axis of symmetry. So the axis of symmetry is if you guys remember.

The formula opposite of b divided by 2a now what was b into what was being a remember all quadratics. All quadratic equations come in the form ax. Squared.

Plus. Bx plus c. Where a b and c are real numbers.

Where a cannot equal 0. So in this formula. I say the axis symmetry or the x coordinate of my vertex is going to equal x equals opposite of b.

Which in this case is a negative 6 divided by 2 times negative 1 where b is negative 6 and a is negative. 1. Anybody having questions up to this point no so therefore i figure this out and i get the x coordinate equals.

3. Right so that means my x. Coordinate is equal to 3.

And my vertex so its three comma. What so ladies and gentlemen in a function. If you know one value if you know that input or the x value how do you find the y value yes.

Shawn you plug it in for your x value right. We know the x we need to figure out the f of x. Or the output value so what i do then is i take f of 3 and i get negative.

3. Squared plus 6 times 3 minus 15. This becomes a negative.

9 plus. 18 minus. 15.

Which ends up giving me f of 3 equals negative. 6. So if i was going to plot this ladies and gentlemen.

I dont really know exactly what this graph looks like but i know right now that i have a point. My vertex is at 3. Negative.

6. 1. 2.

3. 1. 2.

3. 4. 5.

6. Ok. So my points right.

There. Now i need to do is determine is it a maximum or a minimum. What is that point is that the maximum or is that the minimum of the graph.

So remember there was a rule. Theres a test we looked at and that test. Said whenever you get a was less than 1.

Im sorry when a was less than 0. That means you had a maximum point. And when a was greater than 0.

That means your vertex was minimal. What was your minimum point so we look at this and we write down. We say all right what it was my a my a is negative 1 correct so is a greater than or less than less than so therefore that means thats the maximum point right.

So my graphs going to look something like this right. Im just going to were not going to work on sketching it right now well do table values later. So my my maximum point looks like that so we could say thats my vertex.

Which is a maximum point. So thats what you guys should have wrote for your maximum point that is my vertex. Which is a maximum point because the graph opens down.

Because of this rule that we worked on then the next thing that got students. Very confused was the domain and range. Remember.

The domain is the set of all x values that are part of your graph. Ladies and gentlemen look at these two parabolas alright and think about any parabola youve ever graphed or looked at these parabolas are going to infinitely just keep on getting wider and wider right as they go up theyre going to keep on expanding as i go down. They just keep on getting wider.

So eventually they are going to encompass every single x value that we have in the number system every single one so therefore the domain for a quadratic. Without any constraints is going to be all real numbers. So your domain is going to go from negative infinity to infinity pick any value to be.

X. Am i giving a value for x. 2is to a value yes.

Its right there give me another one 18 probably be over here right. But when you go down will this graph eventually have a coordinate at 18 yes right and you can do negative 300 way over there. But its still eventually all the way down there eventually its going to have a point correct.

Because these graphs keep on expanding. However. The range is going to be the y values.

So you look at where are the coordinate points overall. The y values do i have a coordinate point well all these negative numbers yeah. But what about the positives nope so when does it stop well the highest y value that i have a coordinate point was my maximum and my maximum was at negative.

6. So my range is my lowest value which was negative infinity to my highest value which is negative 6. I dont have a y coordinate higher than negative.

6 right what about when y equals zero is that a point on this graph no the highest y value is when y equals negative. 6. So you go to the lowest to your max does anybody have any questions and what i just did no okay.

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