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J H.

In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s). $ f(x) = x^2 + 7 $

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Fill in the blanks. A polynomial function of degree and leading coefficient $ a_n $ is a function of the form $ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 (a_n \neq 0) $ where $ n $ is a ________ _________ and $ a_n, a_{n-1}, \cdots , a_1, a_0 $ are ________ numbers.

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In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s). $ g(x) = x^2 - 8 $

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In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] $ f(x) = 4 - (x - 2)^2 $

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Okay, So in this video, we are going to be talking about in verses of matrices. So within process of matrices, the first thing that you have to know is that it depends on the determinant. So if you haven't seen the determinant, Siri's yet Be sure to go and check that out, because you will need that for the to calculate the inverse. Um And so the reason why is because we know that for the inverse, we basically have to take one over the determinant and multiply that by a different matrix. So in this video, we're mainly going to be concerned with finding the determine are the inverse of a two by two matrix because typically within the scope of pre call classes, you won't have to find the adverse off a three by three because that becomes really complicated. Um, and so you'll mainly find the inter verses of to buy tree matrices. So then the first thing that you have to check out is whether there even is an inverse. So it's very possible for the inverse not to exist because of this component right here. One over the determinant. Because if we have one over the term, the determinant that becomes a fraction. And we know that there are limitations to factions and that you cannot have a zero for the denominator. So if the determinant become zero that it becomes zero in verse, the inverse does not exist again. The reason why is because if we have 1/0 as a fraction, then that entire fraction becomes undefined, which is why we're not allowed to have zero for the denominator of affection, which means that the determinant cannot equal zero. So the big rule that you have to remember here is that the determinant and not equal zero. So then, once you've made sure that the determinant is not equal to zero, so otherwise it could be really any number, whether that's positive or negative, whether that's really big, really small, a fraction and decimal and integer, whatever it is, it's fine. ASL, long as determined, is not zero. Once you've made sure that the inverse does exist, you would multiply the reciprocal of the determinant like so by a particular two by two matrix two by two. And for this, if we were basing it off of our A B. C D two by two matrix. What you would want to do is take this cross and switch the order. So now our D is going to go in the first location and the A is going to go into the in the last location. So then for the B and C, you don't switch it, but you make both of them negative. So it's negative. Be negative. C So then from here, once you've done that, you were just multiplied across because this value right here, the reciprocal of the Dinamo determinant so reciprocal of the determinants um it's just going to become a constant so constant times A matrix is just a coefficient. Times of matrix, which we've learned how to dio. And all you would do from that is just to make sure that you are distributing that coefficient to each position in that matrix. And that's really all you do and you just simplify it and it should become its own two by two matrix. So you'll notice that that two by two matrix is inverse is also going to be a two by two matrix. So that's something to keep in mind if you're ending up with any and any combination that's other than to buy two. Then you probably did something wrong because whatever. That whatever art road times column situation you started with should be what you with for your inverse And just to recap the steps, the first thing is to find the determinant and then take the reciprocal of that. And the second thing is to set up that's matrix, which is the d a. Like you switch the order on. Then do you make the BNC negative? The third thing is just multiplied by that coefficient. In this situation, the coefficient is just that reciprocal of the denominator or determinant. And then the last thing you do is just simplify and evaluate and you should be left with another two by two matrix. So then, one thing that you have to know in terms of once you found your inverse like, how do we check if we've done that? Right. Well, in order to check, we should have our original matrix, our original two by two matrix. And if we multiply that by our inverse, we should end up with a two by two matrix that looks like this. We should have 100 and one. If you multiply it out and you get anything other than that, you probably most likely it was like a error with, like, you're evaluating like your arithmetic. Maybe you messed up a positive for a negative or something like that s o go back and check your work, and you should end up with this two by two matrix of 1001 And that's how you know you did it correctly. So they keep in mind that this order has to be consistent. So your first matric first one has to be the matrix. And the second one has to be your inverse because otherwise you're not gonna get that. So the order matters when we're multiplying this

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