#### _Anonymous

##### Member

- Joined
- Jun 30, 2017

- Messages
- 130

- Gender
- Male

- HSC
- 2019

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".

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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