Transformations Of Radical Equations

Solve equations with radicals and verify extraneous solutions. Grade 12 Math 30-1.

Lesson 6.2 of Radical & Rational Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Parent Function & Parameters. Start with the parent function y = √x and understand its key properties: domain, range, and endpoint. Learn the general form y = a√(b(x - h)) + k and what each parameter controls. Discover the endpoint shortcut and the golden SRT order for applying transformations correctly.
• Sketching Transformed Graphs — Worked Examples. Master the 7-step method for sketching any transformed radical function. Work through detailed examples including y = 3√(x - 1) + 4 and the more complex f(x) = -2√(-¼(x + 1)) - 3 with reflections in both axes. Learn how to identify parameters, find restrictions, map points, and determine domain and range.
• Reading Graphs & Equivalent Forms. Learn to write equations from graphs using endpoint and additional points. Discover the relationship between a-form and b-form equations and prove they're equivalent. Practice converting between forms and understanding how reflections affect the equation. Master the skill of reading transformations directly from any radical function equation.
• Building Equations from Transformations & Applications. Learn to construct equations from verbal descriptions of transformations. Apply radical functions to real-world scenarios like cost of production. Understand how to find all parameters (a, b, h, k) from a list of transformations and interpret mathematical results in context. Practice with domain, range, and y-intercept interpretations in real-world problems.

Inside the lesson — a free preview

What is a Radical Function?

A radical function is any function where the variable appears inside a radical sign. The key requirement is that the variable must be inside the radical — not outside it.

✓ These ARE radical functions:. f(x) = √(5x) and f(x) = 3√(x - 4)

✗ This is NOT a radical function:. f(x) = x√(5) — here the variable x is outside the radical. √(5) is just a constant number.

The Parent Function: y = √(x)

Everything in this lesson builds from one simple function: y = √(x). This is the "parent" — the simplest, unmodified version of a square root function.

Key facts about y = √(x):. • Restriction on radicand: x ≥ 0

• Domain: x ≥ 0

• Range: y ≥ 0

• Endpoint: (0, 0) — this is where the curve begins

Worked example: a: y = 3√(x - 1) + 4

Step 1 — Identify parameters:. a = 3, b = 1, h = 1, k = 4

Step 2 — Restriction on radicand:. x - 1 ≥ 0 → x ≥ 1

Step 3 — Endpoint:. (h, k) = (1, 4)

Step 4 — Mapping rule:. Combine all parameters: (x, y) → (x + h, ay + k) = (x + 1, 3y + 4)

(Since b = 1, there is no horizontal stretch to apply.)

Step 5 — Apply mapping to key points from y = √(x):. We'll map three points from the parent function:

Step 6 — Plot and sketch:. Start at (1, 4) and draw a curve rising to the right through the mapped points.

Step 7 — Domain and Range:. Domain: x ≥ 1 (curve starts at x = 1, goes right)

Range: y ≥ 4 (curve starts at y = 4, goes up)

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