Confusion over function/graph question A Level

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frogstuga
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I don't understand how to finish off this question.

I've worked out the equation of the line being y = 6x + 25, but I'm not quite sure what to do next. I imagine the next step is finding the equation of the parabola. Would this just be (x+2)^2 + 13, or do I have to assume an unknown coefficient?Name:  Screenshot 2021-05-09 at 18.52.36.png
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mqb2766
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Does the quadratic pass through (0,25)?
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Zelerate
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Given the turning point, you get y = (x+2)^2 + 13, but the intercept of this graph would be 17, so you would have to scale the (x+2)^2 term by a coefficient such that the intercept becomes 25.
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deskochan
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It should y = a(x+2)^2 + 13 and then get the value of a from the y-intercept of graph.
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frogstuga
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(Original post by deskochan)
It should y = a(x+2)^2 + 13 and then get the value of a from the y-intercept of graph.
Ahhhhhhh that makes sense now and I see the connection. For the inequality, am I supposed to show that the area is above the quadratic between the points and below the straight line?
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mqb2766
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(Original post by frogstuga)
Ahhhhhhh that makes sense now and I see the connection. For the inequality, am I supposed to show that the area is above the quadratic between the points and below the straight line?
Yes. It needs to satisfy both inequalities. There is no need to mention the points, obviously.
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frogstuga
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(Original post by mqb2766)
Yes. It needs to satisfy both inequalities. There is no need to mention the points, obviously.
How would I go about doing it? I've never seen a question asking something like this so I'm thoroughly confused even though I'll probably kick myself.
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mqb2766
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(Original post by frogstuga)
How would I go about doing it? I've never seen a question asking something like this so I'm thoroughly confused even though I'll probably kick myself.
Just write down the two inequalities for
* above the quadratic
* below the lilne
R is then the intersection of the two. This talks about lilnear inequalities.
https://www.youtube.com/watch?v=rq4N...l=MathswithJay
There are probably better videos/pdfs (textbook?) to describe an inequality.
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mark1666
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If i have understood the question correctly. It is looking for the area of R.


I would do this by integration to find the area below the curve C between the points x=0 and x = -2.

To find this line you would use y = a(x - b)2 + c where (b,c) is the turning point.

In your case it would be:
y = a(x+2)2+13
y = a(x2 +4x + 4) + 13
Using the y intercept to solve for a
25 = a(0 + 0 + 4) + 13
12 = 4a
a = 3
Therefore the curve is y = 3x2 +12x +25

Explained well here
https://www.google.com/search?client...V1fAPpquk8Ac25

I would then work out the area of what is below the line L for the same points. This can be done with either integration or just area of a triangle.


Then subtract the area below the curve from the area below the line. This should give you R

Sorry, I am not going to go through the integration.
Hope that helps.
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deskochan
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This question is really interesting because it needs us to express the answers in a set of inequalities.
This makes me recall the linear programming and graph an area where inequalities overlap. So, what should we do?
we should write all the constraints to enclose the area R.
e.g. y = 6x +25; think about the put (0,0) into it and we can see 0 = 25, ie. 0<25. Yeah, we approach it as y<= 6x +25
then how about parabola? how about x-axis and y axis boundries?
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mqb2766
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(Original post by mark1666)
If i have understood the question correctly. It is looking for the area of R.
....
Could you pls have a read of the sticky at the top of the forum about providing hints, not solutions. Thanks.
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frogstuga
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(Original post by mark1666)
If i have understood the question correctly. It is looking for the area of R.


I would do this by integration to find the area below the curve C between the points x=0 and x = -2.

To find this line you would use y = a(x - b)2 + c where (b,c) is the turning point.

In your case it would be:
y = a(x+2)2+13
y = a(x2 +4x + 4) + 13
Using the y intercept to solve for a
25 = a(0 + 0 + 4) + 13
12 = 4a
a = 3
Therefore the curve is y = 3x2 +12x +25

Explained well here
https://www.google.com/search?client...V1fAPpquk8Ac25

I would then work out the area of what is below the line L for the same points. This can be done with either integration or just area of a triangle.


Then subtract the area below the curve from the area below the line. This should give you R

Sorry, I am not going to go through the integration.
Hope that helps.
Thank you for the steps involving finding a but I already worked it out

That's a great help though thank you very much.
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frogstuga
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(Original post by deskochan)
This question is really interesting because it needs us to express the answers in a set of inequalities.
This makes me recall the linear programming and graph an area where inequalities overlap. So, what should we do?
we should write all the constraints to enclose the area R.
e.g. y = 6x +25; think about the put (0,0) into it and we can see 0 = 25, ie. 0<25. Yeah, we approach it as y<= 6x +25
then how about parabola? how about x-axis and y axis boundries?
That's the issue I'm having. I'm sure the actual inequality is relatively simple and will make me kick myself, but I've just never seen a question asking me to show it in terms of inequalities.
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mqb2766
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(Original post by frogstuga)
That's the issue I'm having. I'm sure the actual inequality is relatively simple and will make me kick myself, but I've just never seen a question asking me to show it in terms of inequalities.
Where does the question come from? Describing inequalities is covered at gcse
https://www.bbc.co.uk/bitesize/guide...qhv/revision/4
I can't see that much a level content in this question
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frogstuga
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(Original post by mqb2766)
Where does the question come from? Describing inequalities is covered at gcse
https://www.bbc.co.uk/bitesize/guide...qhv/revision/4
I can't see that much a level content in this question
This is from the Edexcel A Level 2021 Assessment Materials. This is definitely A Level content as we covered regions like these in Year 12.
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mqb2766
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(Original post by frogstuga)
This is from the Edexcel A Level 2021 Assessment Materials. This is definitely A Level content as we covered regions like these in Year 12.
Just saying the material necessary to answer the question is largely covered at gcse (completing the square, equation of a line, graph inequalities). There is a bit more "higher" content in terms of estimating the quadratic parameters from the points, but not a lot. So yes, A level, but not by much. You could get inequality region description questions at gcse, or when a quadratic curve is less than a line (quadratic iniequalities - gcse), or as the other poster notes, its a fundamental in linear programming (A level option), ... so it shouldn't be too surprising.
https://www.youtube.com/watch?v=aexv...l=corbettmaths
https://www.youtube.com/watch?v=8J_m...l=corbettmaths
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frogstuga
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(Original post by mqb2766)
Just saying the material necessary to answer the question is largely covered at gcse (completing the square, equation of a line, graph inequalities). There is a bit more "higher" content in terms of estimating the quadratic parameters from the points, but not a lot. So yes, A level, but not by much. You could get inequality region description questions at gcse, or when a quadratic curve is less than a line (quadratic iniequalities - gcse), or as the other poster notes, its a fundamental in linear programming (A level option), ... so it shouldn't be too surprising.
https://www.youtube.com/watch?v=aexv...l=corbettmaths
https://www.youtube.com/watch?v=8J_m...l=corbettmaths
Yeah that's true. Well thank you for the all the help anyways I appreciate it. Care to help me on another question I've just posted?
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