Meaning before procedure
For f(x) = a√(x − h) + k, the domain begins at x = h. The range begins at k and goes upward when a > 0 or downward when a < 0. Intercepts are found by setting x or y equal to zero while respecting the domain.
- Domain boundary
- x = h, where the radicand first becomes zero.
- Range direction
- Up for a > 0; down for a < 0.
- x-intercept
- A point where y = 0.
- y-intercept
- A point where x = 0, if 0 is in the domain.
Interactive visual model
Track how the endpoint and the sign of a control domain and range.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Find h from the radicand.
- Use the sign of a to determine range direction.
- Set x = 0 for a possible y-intercept.
- Set y = 0 for x-intercepts and reject inputs outside the domain.
Watch the reasoning, not just the answer
Example 1: Find domain and range of y = 3√(x − 2) − 4.
Example 2: Find the y-intercept of y = √(x + 9) − 1.
Example 3: Find the x-intercept of y = √(x − 1) − 3.
Predict, decide, check, and revise
Build a valid reasoning path
Choose the decisions in the order a careful solver would use them for this concept.
The reasoning path changes as each decision is selected.
Check the foundation
State domain and range of y = 1√(x − (2)) + (-4).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 3√(x − (-1)) + (2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Apply the method with support
State domain and range of y = -1√(x − (4)) + (5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = -2√(x − (-3)) + (1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 2√(x − (0)) + (-2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Work without step prompts
State domain and range of y = 1√(x − (5)) + (3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = -3√(x − (2)) + (-1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 4√(x − (-4)) + (0).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 1√(x − (1)) + (1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Demonstrate durable understanding
State domain and range of y = -1√(x − (-2)) + (-5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 2√(x − (6)) + (-3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = -2√(x − (3)) + (4).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = 5√(x − (-5)) + (2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
State domain and range of y = -4√(x − (0)) + (6).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Analyze y = −2√(x + 4) + 6: endpoint, domain, range, and both intercepts.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Endpoint (−4,6); domain x ≥ −4; range y ≤ 6. Setting x = 0 gives y = 2, so y-intercept (0,2). Setting y = 0 gives √(x+4)=3, so x = 5 and x-intercept (5,0).
The learner record reveals exact answers and model reasoning only after an attempt.