Question: An oceanographer models the salinity of a water sample at depth $ x $ (in meters) with a quadratic polynomial $ s(x) $ such that $ s(1) = 34 $, $ s(2) = 36 $, and $ s(3) = 42 $. Find $ s(0) $. - Parker Core Knowledge

Understanding the Context

Recent interest in ocean salinity models aligns with broader public and scientific attention to climate dynamics, ocean health, and data-informed environmental stewardship. Salinity influences current patterns, marine ecosystems, and global water cycles—increasingly studied in climate adaptation planning. Plotting salinity changes across depth using quadratic functions enables scientists to visualize underlying patterns in complex marine systems. This relevance makes the question resonate with readers exploring climate science, environmental careers, or data-driven research—especially those seeking accessible yet authoritative explanations.

How This Quadratic Model Actually Works

The salinity function $ s(x) = ax^2 + bx + c $ is defined over depth $ x $ in meters. Three data points—s(1) = 34, s(2) = 36, s(3) = 42—provide the foundation for solving for the coefficients $ a $, $ b $, and $ c $. By substituting these values into the polynomial, a system of equations emerges:

• $ s(1) = a(1)^2 + b(1) + c = 34 $ → $ a + b + c = 34 $
• $ s(2) = a(2)^2 + b(2) + c = 36 $ → $ 4a + 2b + c = 36 $
• $ s(3) = a(3)^2 + b(3) + c = 42 $ → $ 9a + 3b + c = 42 $

Key Insights

This system captures how salinity increases with depth, influenced by mixing, temperature, and water movement. Solving these equations reveals the true form of the polynomial and supports predictions beyond the measured points.

Using algebraic elimination, subtract equations to eliminate $ c $:

• (4a + 2b + c) – (a + b + c) = 36 – 34 → $ 3a + b = 2 $
• (9a + 3b + c) – (4a + 2b + c) = 42 – 36 → $ 5a + b = 6 $

Now subtract:
(5a + b) – (3a + b) = 6 – 2 → $ 2a = 4 $ → $ a = 2 $
Substitute $ a = 2 $ into $ 3a + b = 2 $: $ 6 + b = 2 $ → $ b = -4 $
Then use $ a + b + c = 34 $: $ 2 – 4 + c = 34 $ → $ c = 36 $

So, $ s(x) = 2x^2 - 4x + 36 $
Evaluating at $ x = 0 $: $ s(0) = 2(0)^2 - 4(0) + 36 = 36 $

This result reveals how salinity starts at 36 parts per thousand (ppt) at the surface and gradually rises toward larger values with depth—a baseline important for monitoring ocean stratification trends.

Final Thoughts

Common Questions About This Salinity Model

Q: How was the polynomial determined from just three points?
A: By setting up a linear system from the values at $ x = 1, 2,