Introduction — Square Root of a Quadratic. Understand how taking the square root of a parabola creates a function with two separate branches. Learn why y = √(x² − 4) only exists for x ≤ −2 or x ≥ 2, explore the algebraic reasoning behind the gap in the middle, and identify invariant points at (±2, 0) and (±√5, 1).
Examples 1 & 2 — Working with Quadratic Functions. Master two complete examples. Example 1: Given only the graph of a downward parabola with vertex (0, 8) and x-intercepts (±4, 0), find invariant points by inspection, sketch the arch shape between −4 ≤ x ≤ 4 with maximum at (0, 2√2), and derive the equation y = √(−½x² + 8). Example 2: Analyze f(x) = 2x² + 3 and g(x) = √(2x² + 3) — discover that g has no domain restriction, no invariant points, and range y ≥ √3.
The Six Cases — What Shapes Can Appear?. Discover the six possible shapes based on where the parabola sits: Case 1 (vertex below, opens up) creates two outward branches; Case 2 (vertex above, opens down) makes a single arch; Case 3 (vertex on axis, opens up) produces a V-shape; Case 4 (vertex on axis, opens down) is just one point; Case 5 (vertex above, opens up) gives a full U-curve; Case 6 (vertex below, opens down) has no graph at all. Includes complete summary table and memory tricks.
Square Root of General Functions. Apply the universal 5-step process to ANY function shape — waves, piecewise, cubics, anything. Learn the one question that matters: where is f(x) ≥ 0? Master finding x-intercepts, scanning for y = 1 invariant points, and sketching smooth radical curves region by region. Two complete examples: a wave function with gaps and a piecewise linear function with smooth arches inside triangular regions.
Inside the lesson — a free preview
From Linear to Quadratic
The Big Idea. In the previous lesson, you took the square root of a linear function — a straight line. Now you'll do the same thing with quadratic functions (parabolas).
The core idea is identical: y = √(f(x)) is only defined where f(x) ≥ 0, and you take the square root of every y-value while keeping x the same.
The difference is that parabolas can be positive in some regions and negative in others in more complex ways — so the shape of y = √(f(x)) becomes more interesting.
Worked example: Starting Example: y = x^2 - 4
The Original Parabola. The parabola y = x^2 - 4 has:
• Vertex: (0, -4)
• Opens: Upward
• x-intercepts: x = -2 and x = 2
Domain of y = x^2 - 4: x R
Range of y = x^2 - 4: y ≥ -4
Finding the Domain of y = √(x^2 - 4). Now consider y = √(x^2 - 4). The restriction is:
x^2 - 4 ≥ 0
Factor:
(x - 2)(x + 2) ≥ 0
This product is non-negative when:
x ≤ -2 or x ≥ 2
This means the square root function only exists outside the interval (-2, 2).
Between x = -2 and x = 2, the parabola is negative — so the square root is undefined there.
Domain and Range. Domain of y = √(x^2 - 4): x ≤ -2 or x ≥ 2
Range of y = √(x^2 - 4): y ≥ 0
Invariant Points. Where f(x) = 0:
x^2 - 4 = 0
x = ± 2
Invariant points: (-2, 0) and (2, 0)
Where f(x) = 1:
x^2 - 4 = 1
x^2 = 5
x = ±√(5) ≈ ± 2.24
Invariant points: (-√(5), 1) and (√(5), 1)
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