Why is (h,k) the 'Point of Inflection' and 'Centre' in Cubics and Circles? (1 Viewer)

_Anonymous

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As the title states, why is (h,k) the POI (Point of Inflection) and Centre in Cubics and Circles? I understand why they're there for Parabolas (have to find the maximum/minimum value), but what's the reasoning behind Circles and Cubics?

For a Cubic in y = a(x-h)^3 +k form, there are no maximum nor minimum values, so how/why does h,k mean it's the POI (Point of Inflection)? Similarly with Circles, (x-h)^2 + (y-k)^2 form the (h,k) are the 'centres' - why/how do they reflect the circles translating their centres (if that makes any sense)?

Say for example a question is like (x+2)^2 + (y-1)^2 = 4
(h,k) would be (-2,1) respectively. That would mean x and y have to be (-2,1), but wouldn't that lead to the equation being (0)^2 + (0)^2 = 4 - which is false? Same with Cubics y = a(x-3)^3 +k , the x would be 3 since it would lead to 3-3 = 0. Why do we make these values become a 0? I get it for Parabolas, but no clue why for Cubics or Circles.

I asked my teacher this and was told "It's just how it is" but I'm still curious as to why it's "how it is".
 
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Thatstudentm9

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this hsc advanced maths stuff
why are u doing it 5.3 doesn't have this as far i know
 

Cena123

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Say for example a question is like (x+2)^2 + (y-1)^2 = 4
(h,k) would be (-2,1) respectively. That would mean x and y have to be (-2,1), but wouldn't that lead to the equation being (0)^2 + (0)^2 = 4 - which is false?
(-2,1) does not lie on the circle. It is the centre of the circle. Therefore LHS does not equal RHS when the points are subbed.
 

_Anonymous

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(-2,1) does not lie on the circle. It is the centre of the circle. Therefore LHS does not equal RHS when the points are subbed.
So when it's the centre, we imagine it being (0,0)? Why/how is it that (h,k) determine the centre in a circle?
 

jjHasm

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So when it's the centre, we imagine it being (0,0)? Why/how is it that (h,k) determine the centre in a circle?
It's good to see you are thinking over details like these. https://en.wikipedia.org/wiki/Circle <-- Click on that, scroll down "Equataions" then "Cartesian co-ordinates". It gives a solid explanation to your question. Try your best to understand. If you don't understand it, then don't dwell on it too much. If you do then kudos to you. Understanding stuff like this deeply, helps alot with later high school maths. It distinguishes you from the rest.
 

1729

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So when it's the centre, we imagine it being (0,0)? Why/how is it that (h,k) determine the centre in a circle?
Because x^2 + y^2 = r^2 is centred at the origin. And to translate the centre h units to the right, we replace x with (x-h) and k units up, we replace y with (y-k), hence why the circle centred (h,k) has equation (x-h)^2 + (y-k)^2 = r^2

As for why we replace x with (x-h) for a horizontal rightward shift of h units, we are actually translating the coordinate system to the left which appears to be a rightward shift for the graph.
 

kawaiipotato

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As the title states, why is (h,k) the POI (Point of Inflection) and Centre in Cubics and Circles? I understand why they're there for Parabolas (have to find the maximum/minimum value), but what's the reasoning behind Circles and Cubics?

For a Cubic in y = a(x-h)^3 +k form, there are no maximum nor minimum values, so how/why does h,k mean it's the POI (Point of Inflection)? Similarly with Circles, (x-h)^2 + (y-k)^2 form the (h,k) are the 'centres' - why/how do they reflect the circles translating their centres (if that makes any sense)?

Say for example a question is like (x+2)^2 + (y-1)^2 = 4
(h,k) would be (-2,1) respectively. That would mean x and y have to be (-2,1), but wouldn't that lead to the equation being (0)^2 + (0)^2 = 4 - which is false? Same with Cubics y = a(x-3)^3 +k , the x would be 3 since it would lead to 3-3 = 0. Why do we make these values become a 0? I get it for Parabolas, but no clue why for Cubics or Circles.

I asked my teacher this and was told "It's just how it is" but I'm still curious as to why it's "how it is".

 
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_Anonymous

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It's good to see you are thinking over details like these. https://en.wikipedia.org/wiki/Circle <-- Click on that, scroll down "Equataions" then "Cartesian co-ordinates". It gives a solid explanation to your question. Try your best to understand. If you don't understand it, then don't dwell on it too much. If you do then kudos to you. Understanding stuff like this deeply, helps alot with later high school maths. It distinguishes you from the rest.
Thanks for the link, think I've got it now. I then found a similar solution to the Circle problem in a MX1 book which helped as well. Do you know any websites explaining for as to why (h,k) is the POI for Cubics?
 
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_Anonymous

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Because x^2 + y^2 = r^2 is centred at the origin. And to translate the centre h units to the right, we replace x with (x-h) and k units up, we replace y with (y-k), hence why the circle centred (h,k) has equation (x-h)^2 + (y-k)^2 = r^2

As for why we replace x with (x-h) for a horizontal rightward shift of h units, we are actually translating the coordinate system to the left which appears to be a rightward shift for the graph.
Thanks for your help. Do you know why (h,k) is the Point of Inflection for Cubics though?
 

InteGrand

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Thanks for your help. Do you know why (h,k) is the Point of Inflection for Cubics though?
Because the origin is a point of inflection for the un-translated cubic (y = x3).

(Similarly, the origin is the centre of the un-translated circle (x2 + y2 = a2), so (h, k) is the centre of the translated circle (x-h)2 + (y-k)2 = a2. In other words, all you're doing is translating things; the "nature" of the corresponding points are the same.)
 

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