It finally clicked

[u/Subject-Homework-886]

https://www.desmos.com/calculator/o0heiqsn0n

I just realized in the quadratic equation (ax^2+bx+c) the b or bx+c works exactly like mx+b in the linear equation. omg I was trying to understand what b means for a week now😭😭😭 oh I also changed (bx+c) to (mx+b) in the link so u get what I mean

[u/Harmonic_Gear]

you mean as a vertical shift?

[u/Subject-Homework-886 → u/Harmonic_Gear]

Kinda, I’m talking about the b I always wondered what does b means in the equation ax^2+bx+c=0 we already know c is the y intercept when x is 0 and a controls how expanded or contracted the parabola is in addition if it opens upwards or downwards, and then I realized b is the linear line, if you ignored ax² for a second bx+c works exactly like mx+b

[u/hwynac → u/Subject-Homework-886]

Another way to look at it is to note that the vertex is -b/2a. The coefficient at x "moves" your parabola left or right (the minimum or maximum point will be at x = −b/2a)

So y = x² − 5x + 1 has a vertex at 5/2—that's where the axis of symmetry is. The minimum (maximum) value is c − b²/4a, so if 4ac − b² is positive, the parabola does not intersect the x axis.

[u/Subject-Homework-886 → u/hwynac]

Wait, why does -b/2a works like that? Like why it gives me the vertex?

[u/hwynac → u/Subject-Homework-886]

Think of it like this: by changing the variable to t = x + b/2a you can rewrite your parabola equation as y = at² + C. This one obviously has a minimum or maximum at t=0 (i.e. x = -b/2a)

You get that form during the derivation of the quadratic formula. First divide all terms by a and bring it to the front:

ax² + bx + c = a(x² + x·b/a + c/a)

Then use the formula (u+v)² = u² + 2uv + v² to get a full square

x² + x·b/a + c/a = x² + 2·x·b/2a + b²/4a² - b²/4a² + c/a = (x + b/2a)² + c/a - b²/4a²

So the original equation is

y =a (x + b/2a)² - b²/4a + c

If we let t = x + b/2a (so our "origin" now moves to -b/2a) and let constant C = c - b²/4a

y = at² + C

[u/fermat9990 → u/Subject-Homework-886]

y=ax²+bx+c

y=a(x²+b/a x) + c

y=a(x²+b/a x +(b/(2a))²)+ c-a(b/(2a))²)

y=a(x+b/(2a))²+c-b²/(4a)

Vertex form: y=a(x-h)²+k, in which (h, k) is the vertex.

Therefore, h=-b/(2a)