Spatial Geometry & Functions · Common Core High School Algebra I (Quadratic Functions & Graphs), AP Precalculus, GCSE Higher Mathematics (Parabolas & Turning Points).

Quadratic Function & Parabola Moving-Axis Visualizer

Conquer one of algebra's biggest hurdles: dynamic extrema on a closed interval [p, q] as the parabolic axis x = -b/(2a) shifts left, inside, and right. Watch real-time min/max competition.

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param_a

param_h

param_k

interval_p

interval_q

Live Derivations & Metrics

vertex_coord

Stationary turning point

interval_min

Global minimum within [p, q]

interval_max

Global maximum within [p, q]

delta_val

Determines number of real roots

Real-Time Formula Derivation

Governing Formula: --

Substitution: --

Regime: --

Worked Examples & Exam Question Deep Dive

Classic Moving-Axis Problem: Min of y = x² - 2tx + 1 on [-1, 2]

Problem: Given f(x) = x² - 2tx + 1 for x ∈ [-1, 2], find the minimum value function g(t) in terms of parameter t.

Analytical Strategy: The parabola opens upward with symmetry axis x = t. For closed interval [-1, 2], classify into 3 cases: (1) t < -1: increasing, min at f(-1); (2) -1 ≤ t ≤ 2: vertex inside, min at f(t) = 1 - t²; (3) t > 2: decreasing, min at f(2).

Step-by-Step Derivation:

Step 1: Complete the square: f(x) = (x - t)² + 1 - t², axis at x = t.
Step 2: Case 1 (t < -1): Axis lies to left; f(x) strictly increases on [-1, 2]; min is f(-1) = 2t + 2.
Step 3: Case 2 (-1 ≤ t ≤ 2): Axis inside interval; min is vertex value f(t) = 1 - t².
Step 4: Case 3 (t > 2): Axis lies to right; f(x) strictly decreases on [-1, 2]; min is f(2) = 5 - 4t.
Step 5: Synthesize piecewise definition of g(t).

⚠️ Common Misconception Pitfall: When maximizing an upward-opening parabola, the maximum only occurs at the farther boundary; the critical threshold is the interval midpoint (p+q)/2.

Student Inquiry Worksheet & Practice Problems

Q1. What are the axis of symmetry and vertex coordinates for y = 2(x - 3)² + 5?

Q2. Find maximum and minimum of y = x² - 4x + 3 on [0, 5].

Frequently Asked Questions & Misconceptions

What is the rationale behind 'add left, subtract right; add up, subtract down' for function transformations?

Replacing x with (x - h) shifts right because each x must be h units larger to produce the same function input; vertical shift simply adds k directly to the output y.