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Grade 12 · Math · Calculus

12th grade Piecewise continuity and the Intermediate Value Theorem worksheet

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Piecewise continuity and the Intermediate Value Theorem

Grade 12 Math Worksheet - Challenge Me

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Circle the best answer, or write your answer on the line.

  1. f(x) = x² for x < 1, ax + b for 1 ≤ x < 3, and x² for x ≥ 3. Ari makes f continuous. As (a, b), what are a and b?

  2. f(x) = x² for x < 2, ax + b for 2 ≤ x < 4, and x² for x ≥ 4. Bo makes f continuous. As (a, b), what are a and b?

  3. f(x) = x² for x < -1, ax + b for -1 ≤ x < 2, and x² for x ≥ 2. Esme makes f continuous. As (a, b), what are a and b?

  4. f(x) = x² for x < 0, ax + b for 0 ≤ x < 3, and x² for x ≥ 3. Jun makes f continuous. As (a, b), what are a and b?

  5. f(0) = 5; f(1) = 8; f(2) = -2; f(3) = -6; f(4) = 3. Luz studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • at least 2 zeros
    • no zero is promised
    • exactly 2 zeros

  6. f(0) = -7; f(1) = 4; f(2) = 6; f(3) = 9; f(4) = 12. Milo studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • exactly 1 zero
    • no zero is promised
    • at least 1 zero

  7. f(0) = -8; f(1) = 3; f(2) = -5; f(3) = 7; f(4) = -9. Noah studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • exactly 4 zeros
    • at least 4 zeros
    • at least 3 zeros

  8. The air is 12 °C at 6 a.m. and 25 °C at noon, changing continuously. Pia asks whether it is ever exactly 20 °C. Which is true?

    • yes, if it changes at a steady rate
    • yes, at some moment it equals 20 °C
    • not promised: 20 °C is outside the range

  9. A drone climbs continuously from 20 m to 80 m in 10 seconds. Ren asks whether it is ever exactly 95 m. Which is true?

    • yes, at some moment it equals 95 m
    • yes, if it changes fast enough
    • not promised: 95 m is outside the range

  10. Soup cools continuously from 90 °C to 40 °C in an hour. Tala asks whether it is ever exactly 60 °C. Which is true?

    • not promised: 60 °C is outside the range
    • yes, at some moment it equals 60 °C
    • yes, if it changes at a steady rate
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12th grade Piecewise continuity and the Intermediate Value Theorem worksheet – printable
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About this worksheet

Piecewise continuity and the Intermediate Value Theorem. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can check a piecewise function at its join and find f at the join.
  • I can choose constants that make a piecewise function continuous.
  • I can use the IVT to promise a zero or a value, and say when it promises nothing.

Standard: none (beyond CCSS)

Words to know

join: The x where a piecewise function changes rule.

Intermediate Value Theorem (IVT): A continuous f on [a, b] hits every height between f(a) and f(b).

sign change: Neighboring values on opposite sides of 0.

Common mistakes

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Thought the join point is missing. One of the pieces always includes it (≥ or ≤).

Checked only one piece at the join. Both pieces must give the same height.

Questions parents ask

What is the most common mistake with piecewise continuity and the intermediate value theorem?

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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