Lesson 01 · Pattern before formula
Quadratic Patterns & Parabolas
Recognize a quadratic relationship before reaching for a formula: its rate of change changes at a constant rate, producing a symmetric parabolic graph.
The learner question
How can a table reveal that a relationship will bend?
A line adds the same amount each step. A quadratic changes by different amounts, but those changes themselves change by a constant amount.
Signature
constant second differences
For equally spaced x-values, a nonzero constant second difference signals a quadratic pattern.
Worked model
Watch x² change from table to graph.
The outputs 4, 1, 0, 1, 4 do not change by one fixed amount. Their first differences are −3, −1, 1, 3, whose differences are constantly 2.
x
-2-1012
y = x²
41014
first Δ
-3-113
second Δ
222
What bends: the slope is not constant. What stays regular: the slope changes by the same amount each step.
vertex (0, 0)
axis x = 0
Linear
constant first Δ
The output changes by one fixed amount for equal x-steps.
Quadratic
constant second Δ
The first change varies, but its change is constant.
Degree
ax² + bx + c, a ≠ 0
The highest nonzero exponent is 2. If a = 0, the quadratic term disappears.
Difference detector
Classify the pattern from its changes.
x
-2-1012
y
30-103
first Δ
-3-113
second Δ
222
Your classification
Boundary case: y = −x² still has constant second differences. The negative sign flips the parabola downward; it does not remove the quadratic pattern.
Check transferFresh cases, same underlying idea
Quadratic Patterns & Parabolas check1 / 3