Meaning before procedure
In y = a√(x − h) + k, the endpoint is (h,k). The parameter a controls vertical stretch/compression and reflection; h shifts horizontally and k shifts vertically.
- h
- Horizontal shift and endpoint x-coordinate.
- k
- Vertical shift and endpoint y-coordinate.
- a
- Vertical scale; a < 0 reflects across the x-axis.
- Anchor input
- Choose x so x − h is a perfect square.
Interactive visual model
Use the live graph to isolate horizontal shift, vertical shift, scale, and reflection.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Read the endpoint (h,k).
- Choose x-values making x − h a perfect square.
- Multiply root outputs by a and add k.
- Plot anchors and draw only across the real domain.
Watch the reasoning, not just the answer
Example 1: Graph y = √(x − 4) + 2.
Example 2: Graph y = −2√x.
Example 3: Interpret y = 1/2√(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
For y = 1√(x − (2)) + (3), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 1√(x − (-2)) + (1), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Apply the method with support
For y = -1√(x − (4)) + (2), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2√(x − (1)) + (-3), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -2√(x − (-1)) + (4), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Work without step prompts
For y = 1√(x − (5)) + (-2), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 3√(x − (0)) + (1), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -1√(x − (3)) + (-4), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2√(x − (-4)) + (2), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Demonstrate durable understanding
For y = 1√(x − (6)) + (0), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -3√(x − (2)) + (5), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 1√(x − (-5)) + (-1), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2√(x − (4)) + (-2), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -2√(x − (0)) + (3), what is the endpoint?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Describe and graph the transformation from y = √x to y = −√(x − 5) + 1.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Shift right 5 to endpoint (5,0), reflect across the x-axis, then shift up 1 to endpoint (5,1). Anchor points include (6,0) and (9,−1). The domain is x ≥ 5.
The learner record reveals exact answers and model reasoning only after an attempt.