Why Does the Algebraic Function $y = \frac {1}{4c}x^2$ Form a Parabola?

Question

so right now I am taking Algebra 2, and we are learning about Conic Sections. We were of course given the functions to form the shapes of each section (circle, ellipse, parabola, hyperbola). I was just wondering why do these functions form these shapes when graphed? My teacher just gave us the functions, but didn't explain this. Could someone shed some light and/or tell me if this is a useless question? (Let me know if you need me to clarify anything, I'm not sure if I explained my question correctly.)

Answer 1

Suppose we have the focus on the $x$-axis as $F(c,0)$ and the directrix $x=-c$.

Then from the definition, a point $P(x,y)$ on the parabola satisfies $$PF=Pd_\perp$$ thus $$\sqrt{(x-c)^2+(y-0)^2}=|x-(-c)|$$ Simplifying, $$y^2=(x+c)^2-(x-c)^2=4cx$$

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Suppose we have the focus on the $y$-axis as $F(0,c)$ and the directrix $y=-c$.

Then from the definition, a point $P(x,y)$ on the parabola satisfies $$PF=Pd_\perp$$ thus $$\sqrt{(y-c)^2+(x-0)^2}=|y-(-c)|$$ Simplifying, $$x^2=(y+c)^2-(y-c)^2=4cy$$ which gives $$y=\frac{1}{4c}x^2$$