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MATH 1150 · Radical expressions · Objective 1.1

Graph Other Radical Functions and Choose a Model

Extend graphing patterns to fourth roots, fifth roots, and other indices.

CLO 1 · Topic 14 of 21
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Build the concept from the beginning

Meaning before procedure

Even-index radical graphs have a domain boundary and one-sided shape; odd-index radical graphs have all-real domains and pass through a central point. The index controls curvature while a, h, and k control transformations.

Even-index family
Endpoint graph with radicand ≥ 0.
Odd-index family
Two-sided graph with all real inputs.
Index effect
Higher indices flatten differently while keeping parity behavior.
Model selection
Choose a radical family from domain and shape evidence.

Interactive visual model

Change the index and observe the enduring even-versus-odd graph structure.

Change one control at a time and describe what changes—and what stays invariant.

A dependable four-step method

Identify whether the index is even or odd.
Find endpoint or center from h and k.
Use perfect nth powers for anchors.
Apply a and connect points only across the allowed domain.
  1. Identify whether the index is even or odd.
  2. Find endpoint or center from h and k.
  3. Use perfect nth powers for anchors.
  4. Apply a and connect points only across the allowed domain.
Three fully explained examples

Watch the reasoning, not just the answer

Example 1: Describe y = ⁴√x.

Name the goalIndex 4 is even.
Choose the next moveUse x ≥ 0 and principal outputs.
Carry out the mathematicsThe graph begins at (0,0), passes through (1,1),(16,2), and stays in the first quadrant.
Check and interpretFourth powers of the y-values reproduce x.

Example 2: Describe y = ⁵√x.

Name the goalIndex 5 is odd.
Choose the next moveUse positive and negative perfect fifth powers.
Carry out the mathematicsThe graph passes through (−32,−2),(0,0),(32,2) with all-real domain and range.
Check and interpretFifth powers verify each anchor.

Example 3: Choose a family for a graph with endpoint (3,−1) extending only right.

Name the goalA one-sided endpoint signals an even root.
Choose the next moveUse h = 3, k = −1.
Carry out the mathematicsA model is y = a·ⁿ√(x−3)−1 for an even n.
Check and interpretThe domain begins at x = 3, matching the evidence.
Interactive practice

Predict, decide, check, and revise

Interactive construction lab

Build a valid reasoning path

Choose the decisions in the order a careful solver would use them for this concept.

Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.

The reasoning path changes as each decision is selected.

Readiness check

Check the foundation

Which radical graph has a one-sided real domain?

Which radical graph has all real inputs?

Not completed yet.
Guided practice

Apply the method with support

What is the domain of y = ⁴√x?

What is the domain of y = ⁵√x?

Which point lies on y = ⁴√x?

Not completed yet.
Independent practice

Work without step prompts

Which point lies on y = ⁵√x?

A graph has an endpoint and extends right. Which family fits?

A graph passes through a center and extends both ways. Which family fits?

For y = ⁴√(x−3)+2, what is the endpoint?

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Mastery check

Demonstrate durable understanding

For y = ⁵√(x+1)−4, what is the center?

What does the index primarily control in a radical graph?

Which model can decrease and still have all-real domain?

Which evidence is strongest for an even-root model?

Which evidence is strongest for an odd-root model?

Not completed yet.
Unfamiliar transfer

Explain, compare, and revise

A graph has center (−2,4), extends in both horizontal directions, and decreases. Propose a radical model and justify it.

Write a first attempt before opening the self-check.

The learner record reveals exact answers and model reasoning only after an attempt.