Math Help - pTr

[pTr]

Yoo, I'm pTr. I am going to become either a music or math educator in colledge so this really is no big deal for me. Ask me any sort of math questions you have here. This is a serious thread meant to help people so no trolling plz.

I've been meaning to post this for awhile but forgot, got reminded by @THE_V01DL0RD , so thx!

 

[Turns_And_Twists2024]

Thanks for making this thread! i am probably gonna need some math help because i am terrible at math and also will be in high school soon

 

[pTr]

Yeah, ofc. I have a passion for helping others.

 

[SpeedCubingLord]

Why are integrals antiderivatives when they are different concepts? their purposes arent "opposites" to me.

thanks for helping bro

 

[ProStar]

Why are integrals antiderivatives when they are different concepts? their purposes arent "opposites" to me.

thanks for helping bro

Because integration reverses the process of differentiation. The derivative of x^2 is 2x, and the integral of 2x is x^2 [+ C]. It’s the “anti” derivative. They may be used in different situations, but they are two sides of the same coin and come from the same root concepts.

A bit more technically accurate and comprehensive explanation:

Integration measures accumulation. Differentiation measures rate of change. Because the rate of change of an accumulated quantity is the original function, integrals can be computed using antiderivatives. This relationship is formalized by the Fundamental Theorem of Calculus.

 

[Xflighenjoyer]

what is the focus and directix of a parabola?

 

[SpeedCubingLord]

what is the focus and directix of a parabola?

parabola go like basektball

 

[J-EZ_Music]

The focus is a fixed point inside a parabola, and the directrix is a fixed line outside it; a parabola is defined as the set of all points that are equidistant from this focus and this directrix, with the vertex being the midpoint between them, lying on the axis of symmetry.

 

[Xflighenjoyer]

thank you, also how do you find the focus based on a given parabola, and how do you find a parabola given a focus and directix

 

[J-EZ_Music]

Vertical parabola​

(x - h)^2 = 4p(y - k)

• Vertex: (h, k)
• Focus: (h, k + p)
• Directrix: y = k - p

If p > 0: opens up
If p < 0: opens down

Horizontal parabola​

(y - k)^2 = 4p(x - h)

• Vertex: (h, k)
• Focus: (h + p, k)
• Directrix: x = h - p

If p > 0: opens right
If p < 0: opens left


Example

• Focus: (0, 4)
• Directrix: y = 0

Vertex: halfway → (0, 2)
p: 222

(x - 0)^2 = 4(2)(y - 2)
x^2 = 8(y - 2)

Example

• Focus: (3, -1)
• Directrix: x = -1

Vertex: halfway → (1, -1)
p: 2

(y + 1)^2 = 8(x - 1)

 

[Xflighenjoyer]

How do I find P? What is it, I leaned it as (ax+h)^2+k, what’s p?

 

[J-EZ_Music]

p is the distance from the vertex to the focus

 

[Xflighenjoyer]

Thank you

 

[J-EZ_Music]

no problem!