Explanation:
I guess we are only talking about quadratic functions/equations here.
the world is not unified regarding what is the standard form of a quadratic equation.
but let's assume your teacher means
standard form
f(x) = y = ax² + bx + c
vertex form
f(x) = y = a(x - h)² + k
so, we are using the provided x, y coordinates to get a, b and c.
3.75 = a×0.125² + b×0.125 + c
3 3/4 = a×(1/8)² + b×(1/8) + c
15/4 = a/64 + b/8 + c
15×16/64 = a/64 + 8b/64 + 64c/64
=> 240 = a + 8b + 64c
4 = a×0.25² + b×0.25 + c
4 = a×(1/4)² + b×(1/4) + c
4 = a/16 + b/4 + c
4×16/16 = a/16 + 4b/16 + 16c/16
=> 64 = a + 4b + 16c
3 = a×0.5² + b×0.5 + c
3 = a×(1/2)² + b×(1/2) + c
3 = a/4 + b/2 + c
3×4/4 = a/4 + 2b/4 + 4c/4
=> 12 = a + 2b + 4c
we have now from the third equation
a = 12 - 2b - 4c
we put this into the second equation
64 = (12 - 2b - 4c) + 4b + 16c = 12 + 2b + 12c
32 = 6 + b + 6c
26 = b + 6c
b = 26 - 6c
and therefore
a = 12 - 2(26 - 6c) - 4c = 12 - 52 + 6c - 4c = 2c - 40
this a and b we use now in the first equation
240 = (2c - 40) + 8(26 - 6c) + 64c =
= 2c - 40 + 208 - 48c + 64c = 18c + 168
72 = 18c
c = 4
b = 26 - 6×4 = 26 - 24 = 2
a = 12 - 2×2 - 4×4 = 12 - 4 - 16 = -8
so, the standard form is
f(x) = y = -8x² + 2x + 4
the vertex form
we get this by transforming the standard form.
e.g. by competing the square :
y - 4 = -8x² + 2x = -8(x² - x×1/4)
y - 4 -8×(1/8)² = -8(x² -x×1/4 + (1/8)²)
y - 4 - 8/64 = -8(x - 1/8)²
y - 32/8 - 1/8 = -8(x - 1/8)²
y - 33/8 = -8(x - 1/8)²
y = -8(x - 1/8)² + 33/8