Question

(0,
–
3)
(1,4)
(4,
–
8)
(7,5)

Answer 1

I’m not sure what you want done with these points. Do you want them graphed, checked for collinearity, used to find a line or polynomial through them, distances, midpoints, or something else?

Here are some quick useful computations — tell me which (if any) you want to go further with.

Points: (0, −3), (1, 4), (4, −8), (7, 5)

Slopes between successive points
- slope from (0, −3) to (1, 4): 7
- slope from (1, 4) to (4, −8): −4
- slope from (4, −8) to (7, 5): 13/3

No three of those consecutive points are collinear (slopes differ).

Distances between successive points (Euclidean)
- d[(0,−3),(1,4)] = √50 = 5√2 ≈ 7.071
- d[(1,4),(4,−8)] = √153 ≈ 12.369
- d[(4,−8),(7,5)] = √178 ≈ 13.341
- d[(7,5),(0,−3)] = √113 ≈ 10.630

Midpoints of successive segments
- midpoint[(0,−3),(1,4)] = (0.5, 0.5)
- midpoint[(1,4),(4,−8)] = (2.5, −2)
- midpoint[(4,−8),(7,5)] = (5.5, −1.5)
- midpoint[(7,5),(0,−3)] = (3.5, 1)

Cubic polynomial passing exactly through all four points
There is a unique cubic y = ax^3 + bx^2 + cx + d through these four points. Solving gives
a = 149/252, b = −719/126, c = 3053/252, d = −3,
or equivalently
y = (149x^3 − 1438x^2 + 3053x − 756)/252.

Tell me which result you want expanded (graph, simplify polynomial, fit a line, etc.).