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Radical Function Graphs

A free College Algebra lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.

Square-root functions begin where the radicand is zero and extend in the direction where the radicand is nonnegative. Transformations move the starting point.

What you'll learn

Why it matters: Square-root growth, distance formulas, and transformed root models all begin at a domain endpoint. The graph makes the restriction visible: the curve starts where the radicand first becomes nonnegative.

Worked example

Problem. For y = sqrt(x - 4), what is the smallest x-value in the domain?

  1. The radicand must be nonnegative.
  2. x - 4 >= 0.
  3. x >= 4.

Answer: 4

Practice problems

1. For y = sqrt(x + 2), what is the smallest x-value in the domain?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For y = sqrt(x + 2), what is the smallest x-value in the domain?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. x + 2 >= 0.
  4. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

2. The graph of y = sqrt(x) starts at...

Choices: (0, 0) · (1, 0) · (0, 1) · (-1, 0)

Show solution
  1. Core Practice: First identify exactly what the question is asking: The graph of y = sqrt(x) starts at...
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. The parent square-root graph starts at the origin.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (0, 0)

3. For y = sqrt(x - 7) + 3, what is the starting x-value?

Show solution
  1. Challenge: First identify exactly what the question is asking: For y = sqrt(x - 7) + 3, what is the starting x-value?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. The x-shift controls the domain endpoint.
  4. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

4. The graph of y = sqrt(x - 5) starts at...

Choices: (5, 0) · (-5, 0) · (0, 5) · (0, -5)

Show solution
  1. Start Point: First identify exactly what the question is asking: The graph of y = sqrt(x - 5) starts at...
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Set x - 5 = 0.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (5, 0)

5. For y = sqrt(x) + 4, what is the starting y-value?

Show solution
  1. Vertical Shifts: First identify exactly what the question is asking: For y = sqrt(x) + 4, what is the starting y-value?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. The parent graph starts at y = 0, then shifts up 4.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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