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Math 30-1 · Worksheets

Mixed review · Functions and transformations

Ten questions of mixed difficulty, covering Polynomial, radical, rational, Transformations. Print it, or work through it on screen — the answer key starts on its own page.

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Mixed review · Functions and transformations

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. What is the domain of y = √x?

    1. a) x ≥ 0
    2. b) all real numbers
    3. c) x > 0
    4. d) x ≤ 0

  2. The point (3, 4) is on y = f(x). Which point is on y = f(−x)?

    1. a) (−3, 4)
    2. b) (4, 3)
    3. c) (−3, −4)
    4. d) (3, −4)

  3. Compared with y = f(x), the graph of y = f(x) + 5 is:

    1. a) translated 5 units right
    2. b) translated 5 units up
    3. c) translated 5 units down
    4. d) translated 5 units left

  4. The point (2, 5) is on the graph of y = f(x). Which point is on the graph of its inverse?

    1. a) (2, −5)
    2. b) (5, 2)
    3. c) (−5, −2)
    4. d) (−2, 5)

  5. The point (6, 3) is on y = f(x). Which point is on y = −2f(½(x + 4))?

    1. a) (16, −6)
    2. b) (8, 6)
    3. c) (−1, −6)
    4. d) (8, −6)

  6. What is the horizontal asymptote of y = (2x + 1)/(x − 3)?

    1. a) y = 0
    2. b) x = 3
    3. c) y = 2
    4. d) y = −1/3

  7. What are the zeros of y = (x − 2)(x + 5)?

    1. a) 2 and −5
    2. b) −2 and 5
    3. c) 2 and 5
    4. d) −10

  8. Which describes the end behaviour of y = x³?

    1. a) up to the left, down to the right
    2. b) down to the left, up to the right
    3. c) up on both ends
    4. d) down on both ends

  9. Which is a factor of P(x) = x³ − 7x + 6?

    1. a) x + 1
    2. b) x + 2
    3. c) x − 1
    4. d) x − 3

  10. When graphing y = √f(x) from y = f(x), the invariant points occur where:

    1. a) y = 0 and y = 1
    2. b) y = 1 only
    3. c) x = 0 and x = 1
    4. d) the graph crosses the y-axis

Answer key · Mixed review · Functions and transformations

Math 30-1 · maddyhelps.com

  1. a) x ≥ 0 — You cannot take the square root of a negative number, but √0 = 0 is fine, so x ≥ 0.
  2. a) (−3, 4) — The negative is inside with x, so only x changes sign: a reflection in the y-axis. (3, −4) would be y = −f(x).
  3. b) translated 5 units up — Adding outside the function adds to every y-value, so the whole graph moves up 5. Vertical shifts go the same way as the sign.
  4. b) (5, 2) — An inverse swaps x and y, so (2, 5) becomes (5, 2).
  5. d) (8, −6) — x: the ½ stretches horizontally by 2, then shift left 4 — 6 × 2 − 4 = 8. y: stretch by 2 and reflect — 3 × (−2) = −6. Stretches come before translations.
  6. c) y = 2 — The top and bottom have the same degree, so the asymptote is the ratio of leading coefficients: 2/1 = 2. x = 3 is the vertical asymptote.
  7. a) 2 and −5 — Set each factor to zero: x − 2 = 0 gives 2 and x + 5 = 0 gives −5. The signs flip from what you see in the brackets.
  8. b) down to the left, up to the right — Odd degree means the ends go opposite ways; a positive leading coefficient means it rises to the right. So it starts low and ends high.
  9. c) x − 1 — Test each: P(1) = 1 − 7 + 6 = 0, so x − 1 is a factor. P(−1), P(3) and P(−2) all give 12, not 0.
  10. a) y = 0 and y = 1 — √0 = 0 and √1 = 1, so points with those y-values stay exactly where they are. Everywhere else the y-value changes.