Question

Write a quadratic function that fits the given set of points.

(2, 3), (6, 3), and (8, -3)

Answer 1

@mathmale

Answer 2

I believe you did exactly the right thing before. Is this the same problem as before, or a new one? I thought we were at the point where we needed to solve a system of linear equations.

Answer 3

The same one!

Answer 4

Here's where you were:

3 = 4a + 2b + c
3 = 36a + 6b + c
-3 = 64a + 8b + c

We were talking about how to solve this system for the constants a, b and c.
I asked you to identify the method you'd prefer to use to do so.
Just say what you want to do, and we'll go ahead with it.

Answer 5

Here are some of the methods you have to choose from:
elimination by addition and subtraction
elim. by substitution
matrix algebra
determinants

Answer 6

I am sorry that it has take so long to reply. My girlfriend called to cheer me up, I have been stressed today, it has been a busy day with church bands and work. But I am ready to get this work done!

The only one of those options I am really familiar with is Elimination by subtraction.

Answer 7

We'll do the problem that way then.
I'm going to modify your set of equations just a little: from:

3 = 4a + 2b + c
3 = 36a + 6b + c
-3 = 64a + 8b + c

4a + 2b + c = 3
36a +6b +c =3
64a+8b+c=-3

Answer 8

All i did was to move the constants to the right side from the left side.

Look at the first 2 equations, Rick: How would you go about eliminating c?

Answer 9

36a +6b +c =3
-(4a + 2b + c = 3)
-----------------
?

Answer 10

Would I subtract one from the other?

4a + 2b + c = 3
- 36a +6b +c =3
---------------

Answer 11

Oh yes! Ha! I was right!

Answer 12

Yes. Please go ahead with it now.

Answer 13

Suggest doing it on paper if you aren't already. Helps when you can see what you've already done. You might want to save this paper and keep such papers in order so that you have something to review later.

Answer 14

Yes, I am doing it on paper, I will have it finished in just a bit.

Answer 15

It would be:

-32a - 4b = 0

Answer 16

That's great. I have the same result, except I chose to use all + signs. Nice work!
Now do exactly the same thing with the 1st and 3rd equations; elim c again. Hold on to your first result; we'll use it later.

Answer 17

So.

4a + 2b + c = 3
- 64a + 8b + c = -3
------------------
?

Answer 18

Yes. try it.

Answer 19

-60a - 6b = 6 ??

Answer 20

Again, perfect! You do know your stuff.

Now y ou have 2 equations in 2 unknowns, with c gone.

Answer 21

Yes!

Answer 22

-60a - 6b = 6
-32a - 4b = 0

Answer 23

How would you go about eliminating either a or b (your choice)?

Answer 24

Divide from both sides.

Answer 25

or would I subtract from bot sides?

Answer 26

Unsure of what you mean. What would the purpose of doing that be? I do see that both equations can be reduced; is that what you meant?

Answer 27

In order to isolate "a", would I subtract -32 or -60 from both sides of the equation?

Answer 28

-60a - 6b = 6 => -10a - b = 1
-32a - 4b = 0 => -8a - b =0

To answer your question (to be continued)

Answer 29

You could, if you wanted, divide the first equation by 60 or -60 or whatever, but subtracting 60 or 32 on both sides would be a mistake.

Answer 30

Oh! I didn't even realize that! Wow, that was literally right in front of me. Okay! So now we have them reduced.

Answer 31

Simply change all the signs in one of the equations on the right:

-60a - 6b = 6 => -10a - b = 1 => 10a + b = -1
-32a - 4b = 0 => -8a - b =0 => -8a -b = 0

Answer 32

Yes, you could reduce them if you want.

But please take a look at this latest set of simultaneous equations. Do you see what to do next? If so, do it, please.

Answer 33

This is "elimination by subtraction."

Answer 34

SO 2a = 1 ?

Answer 35

Check that. One small error.

Answer 36

hmm.

10a + b = - 1

Answer 37

so it would be

2a = -1

Answer 38

2a = 1 should be 2a = -1. Agreed?

Answer 39

yes, you had it correct before my question got posted!

Answer 40

solve that for a, please

Answer 41

a = -1/2

Answer 42

Perfect. As I said earlier, I'd already solved this system using matrices on my TI-83 calculator. I also got a=-1/2. Nice work! Now go back to:

Answer 43

10a + b = -1 and substitute a=-1/2, to find b. b=?

Answer 44

b = 4

Answer 45

Perfect!
Now with b=4 and a=-1/2, go back to any one of the 3 original equations and find c.

Answer 46

would c = -3 ?

Answer 47

Again, right on! So, your quadratic becomes
y=-0.5x^2 + 4x - 3

Answer 48

Please take a look at its graph:

https://www.google.com/search?q=y%3D-.5x%5E2%2B4x-3&rlz=1C1CHFX_enUS461US461&oq=y%3D-.5x%5E2%2B4x-3&aqs=chrome..69i57j0l5.16314j0j7&sourceid=chrome&espv=210&es_sm=122&ie=UTF-8

Answer 49

going back to the original question, where you were given 3 points, you'll see that 2 of the points were (2,3) and (6,3).

Look at the graph. Do these points seem to satisfy the equation (do they lie on the graph)?

Answer 50

Awesome! And all of that only took up half of a page! I really wish I had a graphing calculator!

Answer 51

And yes, they do fit the graph!

Answer 52

yes, I'd recommend you get one if you're taking much more math. We could talk about this later.
So you see, you have solved this problem completely.

Starting from 3 points,

Answer 53

you've come up with the equation of a quadratic function satisfied by the points.

Answer 54

Okay! so now I have that, I can do it and I took good notes!

I am going to show you some other things that have confused me if that is alright! I can finish the rest of these later

Answer 55

Hope you have some way of capturing what we've typed here; it would be tremendously useful to you to have something to review. I know from experience that once I've learned just so much, my mind cannot absorb any more for a while, at least.

Answer 56

Great minds think alike, as they say. :)

Answer 57

Ready, aim, fire questions. Then choose a new problem on which to focus.

Answer 58

got your photo; waiting for your questions.

Answer 59

This is the question, there are two like this, I don't quite understand.

Answer 60

I have to find a quadratic, then I have to estimate how much the cost would be for 24 in. x 36 in. photo

Answer 61

image very slow in loading; let me check it again

Answer 62

got image, looking at it now

Answer 63

If I read this correctly, you'll need to average 24 and 36. Bet you've done that before.

Answer 64

average them? Yes.

Answer 65

my comment was out of order; sorry.

Yes, you're right, we have to come up with a quadratic model just as we did in the last problem.

Answer 66

If we average them, the mean would be 30

Answer 67

Your table shows a bunch of photo sizes.
Find the average of each set of dimensions. For exampel, for the 8 by 10, the average of the 2 given dimensions is (8+10)/2=9.
Please go ahead and find the averages of the other sets of dimensions.
We'll need this for ffidning the quadratic model.

Answer 68

Oh! okay! yes I will do that. And I have a question. When we find the averages, do we have to use every plot point or just three like we did the last?

Answer 69

@mathmale

Answer 70

I'd suggest you do it for all of the table. we won't need that may points, however. As before, to find a quadratic model, we'll need just 3 points (you're right).

Answer 71

Let's compare results.

I've obtained:

9 10
16 16
18 19
27 27
36 39

Answer 72

that was very insightful of you.

Answer 73

Yes! that is exactly what i have, I have already started working on finding the quadratic. I am about to solve for a

Answer 74

Yes, use the same quadratic model you started with before: y=ax^2+bx+c.
choose any three of the points listed and write 3 equations, just as you did with the last problem.

Answer 75

Once you have those 3 equations, you can solve them in the same way you did the previous system of linear equations. I'd prefer to move on to a new problem, but would be glad to go through the solution of this present system of linear equations if you like.

Answer 76

Excuse me for a moment; I need to make a brief phone call. I'll be back!

Answer 77

You don't have to type your 3 equations here unless you want to go through the solution with me.

Answer 78

I am going to solve it and show you, I have a good feeling about this!

Answer 79

OK. Just wanted to let you know that I'm back. Also delighted that you have a good feeling about your ability to solve this kind of problem!

Answer 80

Okay, so I am trying to solve, but I have hit a problem. I subtracted the first and second equations and the first and third equations. But now I am stuck at:


-156a - 6b = -7
-224a - 8b = -10

Answer 81

I don't know whether or not that is correct as I haven't attempted the problem myself, but your results here do look appropriate.

Answer 82

would you like me to take a picture of my work?

Answer 83

It will be a little harder to solve this system because the 6 and the 8 coefficients of b are different; do you know what to do in a case like this or should we discuss what to do?

Answer 84

I would like to discuss it please! Because I have no clue of what to do.

Answer 85

Coming up.

Answer 86

Would we isolate "a" ?

Answer 87

In
-156a - 6b = -7
-224a - 8b = -10
we want to eliminate either a or b, right? b would be easier to elim.

Answer 88

Yes, elim. b would give us an expression in a which would be easy to solve for a.
Let's keep the 1st eq'n as is.
We want to change the 2nd eq'n so that the coeff. of b is +6; then -6b + 6b = 0 as desired.
Right?

Answer 89

Yes!

Answer 90

Here's what I propose. Just multiply everythign in the 2nd row by (-6/8). Just do it, and then perhaps the reason for doing this will become apparent.

Answer 91

Okay, would the first equation now become -156a - b = -1 ?

Answer 92

And what do you mean by "multiply everything in the 2nd row by (-6/8)"?

Answer 93

Okay, would the first equation now become -156a - b = -1 ? NO.
The second row is -224a - 8b = -10.
Multiplying everything in this row by (-6/8) results in 168a + 6b = 7.5. Please look this over and see whether you can agree with that or not.

Answer 94

So where did the (-6/8) come from exactly?

Answer 95

Now, the first row is still -156a - 6b = -7 I asked you to leave that as it is.
The second row (new) is 168a + 6b = 7.5

Is that what y ou got? If so, add the like terms together (which will eliminate b).
Mind if we finish this first, and then I'll answer your qu. about where that -6/8 came from.

Answer 96

Yes! okay, so now I have 12a = 1/2