Lesson 57: Graphing Radical Functions
Radical functions include variables under a radical, such as square roots or cube roots. Their graphs have characteristic shapes, domains, and ranges. Understanding transformations helps you sketch them quickly.
Key Points
- Square root function: f(x) = √x → domain: x ≥ 0, range: y ≥ 0, shape: increasing curve starting at origin.
- Cube root function: f(x) = ∛x → domain: all real numbers, range: all real numbers, shape: increasing S-curve through origin.
- Transformations: vertical/horizontal shifts, reflections, and stretches/compressions affect the graph.
Examples
Example 1: f(x) = √(x − 3)
- Shift the basic √x graph 3 units to the right
- Domain: x ≥ 3, Range: y ≥ 0
Example 2: g(x) = −√x + 2
- Reflect √x over x-axis and shift up 2 units
- Domain: x ≥ 0, Range: y ≤ 2
Example 3: h(x) = ∛(x − 1)
- Shift ∛x graph 1 unit to the right
- Domain and Range: all real numbers
Practice Problems
- Graph f(x) = √(x + 4)
- Graph g(x) = √x − 5
- Graph h(x) = −√(x − 2)
- Graph k(x) = ∛(x + 3)
- Graph m(x) = 2√(x − 1) + 1