The square root function
The square root function f(x) = √x is the basic radical function.
| x | 0 | 1 | 4 | 9 | 16 |
|---|---|---|---|---|---|
| √x | 0 | 1 | 2 | 3 | 4 |
Graph: starts at the endpoint (0, 0) and curves upward to the right, flattening as it goes.
- Domain: x ≥ 0
- Range: y ≥ 0
It’s half of a sideways parabola: √x is the inverse of y = x² for x ≥ 0.
Transformations
y = a√(b(x − h)) + k
| Parameter | Transformation |
|---|---|
| h | horizontal shift (right if h > 0) |
| k | vertical shift |
| a | vertical stretch; reflection in the x-axis if a < 0 |
| b | horizontal stretch; reflection in the y-axis if b < 0 |
The endpoint moves to (h, k).
Examples
| Function | Endpoint | Direction | Domain | Range |
|---|---|---|---|---|
| y = √(x − 3) | (3, 0) | right, up | x ≥ 3 | y ≥ 0 |
| y = √x + 2 | (0, 2) | right, up | x ≥ 0 | y ≥ 2 |
| y = −√x | (0, 0) | right, down | x ≥ 0 | y ≤ 0 |
| y = √(−x) | (0, 0) | left, up | x ≤ 0 | y ≥ 0 |
| y = 2√(x + 1) − 4 | (−1, −4) | right, up | x ≥ −1 | y ≥ −4 |
Finding the domain
Set the radicand ≥ 0.
- f(x) = √(2x − 6) → 2x − 6 ≥ 0 → x ≥ 3
- f(x) = √(5 − x) → x ≤ 5
Cube root functions
f(x) = ∛x is defined for all real numbers (cube roots of negatives exist). Its graph passes through (−8, −2), (−1, −1), (0, 0), (1, 1), (8, 2) and has an S-shape.
Graphing y = 2√(x + 1) − 4
- Endpoint (−1, −4).
- Base points of √x: (0, 0), (1, 1), (4, 2), (9, 3).
- Multiply y by 2, then shift left 1 and down 4: (−1, −4), (0, −2), (3, 0), (8, 2).
- Draw the curve.
Applications
- Pendulum period: T = 2π√(L/g)
- Speed from skid marks: s = √(30fd) (an empirical formula in miles per hour, used in accident reconstruction)
- Falling time: t = √(2h/g)
- Side of a square from its area: s = √A
Worked examples
Example 1: Domain and range of y = √(x + 4) − 1? x ≥ −4, y ≥ −1.
Example 2: Endpoint of y = −3√(x − 2) + 5? (2, 5), curving down.
Example 3: Domain of y = ∛(x − 7)? All real numbers.
Common mistakes
- Shifting the wrong way: √(x − 3) moves right, not left.
- Forgetting reflections change the range.
- Restricting the domain of cube roots.
Try it yourself
- Domain of f(x) = √(x − 9)?
- Range of f(x) = −√x + 6?
- Endpoint of y = √(x + 2) + 3?
- Domain of f(x) = √(10 − 2x)?
Answers: 1) x ≥ 9 2) y ≤ 6 3) (−2, 3) 4) x ≤ 5
Quick recap
- The square root function starts at an endpoint and curves.
- Domain: radicand ≥ 0; range depends on reflections and shifts.
- Use transformations of √x to graph.