this hsc advanced maths stuff
why are u doing it 5.3 doesn't have this as far i know
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".
Last edited by _Anonymous; 14 Oct 2017 at 9:32 PM.
this hsc advanced maths stuff
why are u doing it 5.3 doesn't have this as far i know
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.
2016: Mathematics (89)
2017: Economics - Biology - Extension 1 Maths - Extension 2 Maths - English (ADV)
2018: Bachelor of Commerce/Bachelor of Science (Advanced Maths) @ UNSW
ATAR Goal: 96+
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.
Last edited by kawaiipotato; 16 Oct 2017 at 1:12 AM.
Sxc avatar made by a sxc person: carrotontheground http://community.boredofstudies.org/...otontheground/
Last edited by _Anonymous; 1 Nov 2017 at 9:09 PM.
Because the origin is a point of inflection for the un-translated cubic (y = x^{3}).
(Similarly, the origin is the centre of the un-translated circle (x^{2} + y^{2} = a^{2}), so (h, k) is the centre of the translated circle (x-h)^{2} + (y-k)^{2} = a^{2}. In other words, all you're doing is translating things; the "nature" of the corresponding points are the same.)
Sxc avatar made by a sxc person: carrotontheground http://community.boredofstudies.org/...otontheground/
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