Meaning before procedure
For y = a∛(x − h) + k, the central anchor is (h,k). The graph keeps an all-real domain and range. The sign and size of a control reflection and vertical scale.
- Center
- The transformed point (h,k).
- Horizontal shift
- Controlled by h inside x − h.
- Vertical shift
- Controlled by k outside the radical.
- Reflection
- Occurs when a < 0.
Interactive visual model
Manipulate the cube-root center, scale, and reflection without creating a false endpoint.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Plot the center (h,k).
- Choose x − h = −8, −1, 0, 1, 8.
- Multiply cube-root outputs by a and add k.
- Connect the anchors as a continuous transformed S-curve.
Watch the reasoning, not just the answer
Example 1: Graph y = ∛(x − 2) + 1.
Example 2: Graph y = −∛x.
Example 3: Interpret y = 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 center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 1∛(x − (-2)) + (1), what is the center?
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 center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2∛(x − (1)) + (-3), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -2∛(x − (-1)) + (4), what is the center?
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 center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 3∛(x − (0)) + (1), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -1∛(x − (3)) + (-4), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2∛(x − (-4)) + (2), what is the center?
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 center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -3∛(x − (2)) + (5), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 1∛(x − (-5)) + (-1), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = 2∛(x − (4)) + (-2), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
For y = -2∛(x − (0)) + (3), what is the center?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Graph y = −2∛(x − 4) + 3 and describe every transformation.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Shift right 4, reflect across the x-axis, stretch vertically by 2, and shift up 3. The center is (4,3); nearby anchors include (3,5) and (5,1). Domain and range are all real numbers.
The learner record reveals exact answers and model reasoning only after an attempt.