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Math 20-1

Math 20-1 · Worksheets

Quadratic functions

Ten questions of mixed difficulty, covering Quadratic functions. Print it, or work through it on screen — the answer key starts on its own page.

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Quadratic functions

Math 20-1 · maddyhelps.com

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

  1. Written in vertex form, y = x² + 6x + 5 is

    1. a) y = (x + 6)² − 31
    2. b) y = (x − 3)² − 4
    3. c) y = (x + 3)² + 5
    4. d) y = (x + 3)² − 4

  2. The graph of y = a(x − p)² + q has no x-intercepts when

    1. a) a > 0 and q < 0
    2. b) a < 0 and q > 0
    3. c) a > 0 and q > 0
    4. d) p = 0

  3. The form that shows the vertex without any extra work is

    1. a) y = a(x − r₁)(x − r₂)
    2. b) y = a(x − p)² + q
    3. c) y = mx + b
    4. d) y = ax² + bx + c

  4. The graph of y = −2(x + 1)² + 4

    1. a) opens downward and has a minimum value of 4
    2. b) opens downward and has a maximum value of 4
    3. c) opens upward and has a minimum value of −2
    4. d) opens upward and has a maximum value of 4

  5. A ball is thrown so that its height is h = −5t² + 20t + 1, with h in metres and t in seconds. Its maximum height is

    1. a) 21 m
    2. b) 20 m
    3. c) 41 m
    4. d) 2 m

  6. The vertex of y = (x − 3)² + 5 is

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

  7. The y-intercept of y = x² + 6x + 5 is

    1. a) 1
    2. b) 5
    3. c) 6
    4. d) −5

  8. The range of y = −3(x − 2)² + 7 is

    1. a) y ≥ 7
    2. b) y ≤ 7
    3. c) y ≤ −3
    4. d) all real numbers

  9. A rectangular pen is built with 40 m of fencing. The greatest area it can enclose is

    1. a) 80 m²
    2. b) 400 m²
    3. c) 100 m²
    4. d) 160 m²

  10. The vertex of y = 2x² − 8x + 3 is

    1. a) (2, −5)
    2. b) (4, 3)
    3. c) (2, 5)
    4. d) (−2, 27)

Answer key · Quadratic functions

Math 20-1 · maddyhelps.com

  1. d) y = (x + 3)² − 4 — Half of 6 is 3, and 3² = 9, so x² + 6x = (x + 3)² − 9. Then y = (x + 3)² − 9 + 5 = (x + 3)² − 4. Check with x = 0: (3)² − 4 = 5 ✓, matching the original constant.
  2. c) a > 0 and q > 0 — An upward parabola whose lowest point is already above the x-axis never reaches it. The same is true the other way: a < 0 with q < 0. What matters is that a and q have the same sign.
  3. b) y = a(x − p)² + q — Vertex form is built so that the vertex is (p, q). Standard form hides it — you have to complete the square or use x = −b/2a. Factored form shows the x-intercepts instead.
  4. b) opens downward and has a maximum value of 4 — The sign of a decides the direction: a = −2 is negative, so the parabola opens downward and the vertex is the highest point. The vertex is (−1, 4), so the maximum value of y is 4.
  5. a) 21 m — The maximum is at the vertex: t = −b/(2a) = −20/(−10) = 2 seconds. Then h = −5(4) + 40 + 1 = 21 m. The value 2 is when it happens, not how high — the question asks for the height, so you have to substitute back.
  6. a) (3, 5) — In y = a(x − p)² + q the vertex is (p, q). The subtraction is built into the form, so x − 3 means p = 3, not −3. The sign flip catches people every time.
  7. b) 5 — The y-intercept is the value when x = 0, which in y = ax² + bx + c is always c. So it is 5. Reading the constant straight off is one of the few things standard form makes easier than vertex form.
  8. b) y ≤ 7 — The parabola opens downward because a is negative, so the vertex value 7 is the largest y ever gets, and everything below is reachable. Range is about y-values; the domain here is all real numbers.
  9. c) 100 m² — With perimeter 40, the sides are x and 20 − x, so A = x(20 − x) = −x² + 20x. The vertex is at x = 10, giving A = 100 m². For a fixed perimeter the square always wins, which is worth remembering as a check.
  10. a) (2, −5) — The axis of symmetry is x = −b/(2a) = 8/4 = 2. Substituting back: y = 2(4) − 16 + 3 = −5. So the vertex is (2, −5). Forgetting the minus sign in −b/(2a) sends you to x = −2.