Please discuss the relationship between the value of "a" in a quadratic function [y = a(x - h)2 + k or y = ax2 + bx + c] and the distance the focus is from the vertex. Please give examples to support your conclusion. Please Help!

The given equations are;

y = a·(x - h)² + k or y = a·x² + b·x + c

We have;

(x - h)² = 4·p·(y - k),

Where;

(h, k) are the x, and y coordinates of the vertex

p = The distance from the focus to vertex distance;

Comparing, we have;

y - k = a·(x - h)²

(y - k)/a = (x - h)²

(x - h)² = 4·p·(y - k)

4·p·(y - k) = (x - h)²

∴ 4·p = 1/a

p = 1/(4·a)

Therefore, the focus to vertex distance inversely proportional to the value of a

The distance of every point on the parabola is equidistant from the directrix and the focus

Therefore, the equation of the directrix is y = k - p

For example, we have;

y = 2·x² + 3·x + 1

h = -b/2a = -3/4

k = (4ac - b²)/4a = (4×2×1 - 3²)/(4×2) = -1/8

The coordinate of the focus is (h, k + p) = (-3/4, -1/8 + 1/8) = (-3/4, 0)

The directrix, y = k - p = -1/8 - 1/8 = -1/4

For a point x = 2, we have, y = 2·2² + 3·2 + 1 = 15

The coordinate of the point is (2, 15)

The distance of the point from the directrix = 15 - (-1/4) = 15 + 1/4 = 15.25

The distance of the point from the focus is √((2 - (-3/4))² + (15 - 0)²) = 15.25

Therefore, all points on the parabola have equal distances from the focus and the directrix

Step-by-step explanation:

there are infinitely many solutions.