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Find the function $y=a x^{2}+b x+c$ whose graph contains the points $(1,2),(-2,-7),$ and (2,-3)....

Question

Find the function $y=a x^{2}+b x+c$ whose graph contains the points $(1,2),(-2,-7),$ and (2,-3).

Find the function $y=a x^{2}+b x+c$ whose graph contains the points $(1,2),(-2,-7),$ and (2,-3).



Answers

The graph of each function contains the given point. Find the value of $c .$
$$
y=2 x^{2}+c ;\left(-\frac{3}{4},-\frac{1}{4}\right)
$$

So here we have. Why ankles to explain? Plus c the condition Negative. 34 Come one for that was fun that it One floor equals two times 3/4 square, See, or native 34 That tube is 9 16 I see. Florist to nine over eight. Risky. So C is equal to 1/4. But in this 98 is equal to two with minus 98 which was equal to Native Said it over eight. Oh, sorry. I take that back. This is negative. 1/4. So I'm gonna put a negative here, making it negative. Negative. Negative too. Service isn't negative. 11 over eight is the answer.

Okay, so we're asked to put, um, these two points into proble equation form. I mean, they tell us that it's a X squared plus c, and we can see here that this is going to be our ver tax because when we have no be term than our accesses cemetery is on the why access. And so here is that point. And so I know that when we have no be term that are Vertex is at zero c. So I know that sea is negative three. So now I've got a negative three in the sea possession And so I have an accent. Why? Using this point over here and then I consol for a So I'll put a negative seven in for Why a and then acts was one and it squared minus three. So add three to both sides. So I get that's negative four, and I get that a is negative four because one squared is just one. And so now I can write that whole equation out as why and then a negative four is our A. And then I have X squared minus three

All right. So this problem here asks us to find the quadratic equation here in the form of y equals X squared plus c using these given points. And so this time we don't have a ver tax point, so that does not give us See. So when we have this many unknowns in an equation and we will know an accident, why you concede that we haven't a NSC left? We just need to equations for that. So we'll just plug in each of the one point and then solve for that. We can use either substitution or elimination. And so what we have here, I'll start with the bigger numbers here because I'm going to use, um, elimination. And so if I plug in 89 for why and that's equal to a an ax squared is negative. Three squared plus c and then I'll use this other point. So I have 39 is equal to a times two squared plus C. And so I chose elimination because if I just subtract each one of these terms from these bottom terms from the top term, then that gets rid of the sea. So that's just kind of Ah, quick and easy way to do that. And so here, 89 minus 39 is 50 and then I have nine A. So this will be nine a, and this will be for a so a nine a minus for a is five a and a C minus c zero. And so now to solve for a I'm just going to divide both sides by five, and I get that a is equal to 10. So now I can plug back into this equation using either one of these points and a as 10 and then I'll only have one unknown the sea, and I'll be all the self receipt. So I'm you know, I love using the smaller numbers. Um, so I'll start with that one and say that 39 is equal to 10 cause that's what the day we found times ax squared. So this is two squared plus c. So here this is a four. And so this becomes 39 is equal toe 40 plus c subtracting 40 from both sides. I get that c is equal to negative one. So putting that all together, I get that why is equal to 10 x squared minus one


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