Quadratic Functions practice questions

Quadratic Functions questions

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1. 1Quadratic functionsStandard

   What is the vertex of y = 2(x - 3)² + 1?What is the vertex of y = 2(x - 3)^2 + 1?

   • A(3, -1)(3, -1)
   • B(3, 1)(3, 1)
   • C(-3, 1)(-3, 1)
   • D(1, 3)(1, 3)
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   Answer: (3, 1)(3, 1)

   In vertex form a(x - h)² + k, the vertex is (h, k). Here h = 3 and k = 1.In vertex form a(x - h)^2 + k, the vertex is (h, k). Here h = 3 and k = 1.

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2. 2Quadratic functionsStandard

   What is the axis of symmetry of the parabola y = x² - 10x + 7?What is the axis of symmetry of the parabola y = x^2 - 10x + 7?

   • Ax = -5x = -5
   • Bx = 7x = 7
   • Cx = 5x = 5
   • Dx = 10x = 10
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   Answer: x = 5x = 5

   The axis of symmetry of y = ax² + bx + c is x = -b / (2a). Here that is --102 × 1 = 5.The axis of symmetry of y = ax^2 + bx + c is x = -b / (2a). Here that is -(-10)/(2 × 1) = 5.

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3. 3Quadratic functionsStandard

   The graph of y = (x - 2)(x + 6) crosses the x-axis at two points. What is the x-coordinate of its vertex?The graph of y = (x - 2)(x + 6) crosses the x-axis at two points. What is the x-coordinate of its vertex?

   • A-6-6
   • B22
   • C-2-2
   • D44
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   Answer: -2-2

   The roots are x = 2 and x = -6, and a parabola is symmetric about the midpoint of its roots: 2 + (-6)2 = -2.The roots are x = 2 and x = -6, and a parabola is symmetric about the midpoint of its roots: (2 + (-6))/2 = -2.

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4. 4Quadratic functionsStandard

   The parabola y = x² + bx + 7 has its vertex at x = 3. What is b?The parabola y = x^2 + bx + 7 has its vertex at x = 3. What is b?

   • A-6-6
   • B66
   • C-3-3
   • D33
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   Answer: -6-6

   The vertex sits at x = -b / (2a) with a = 1, so -b / 2 = 3 and b = -6.The vertex sits at x = -b / (2a) with a = 1, so -b / 2 = 3 and b = -6.

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5. 5Quadratic functionsStandard

   At which points does the graph of y = x² - 8x + 15 cross the x-axis?At which points does the graph of y = x^2 - 8x + 15 cross the x-axis?

   • Ax = 3 and x = 5x = 3 and x = 5
   • Bx = -3 and x = -5x = -3 and x = -5
   • Cx = 15 and x = 1x = 15 and x = 1
   • Dx = 8 and x = 15x = 8 and x = 15
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   Answer: x = 3 and x = 5x = 3 and x = 5

   The graph crosses where y = 0, and x² - 8x + 15 factors as (x - 3)(x - 5). The signs in the brackets are the opposite of the roots, which is what the second choice gets wrong.The graph crosses where y = 0, and x^2 - 8x + 15 factors as (x - 3)(x - 5). The signs in the brackets are the opposite of the roots, which is what the second choice gets wrong.

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6. 6Quadratic functionsChallenge

   What is the minimum value of y = x² - 6x + 11?What is the minimum value of y = x^2 - 6x + 11?

   • A33
   • B1111
   • C-2-2
   • D22
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   Answer: 22

   Complete the square: y = (x - 3)² + 2. The squared term is never negative, so the smallest value of y is 2, reached at x = 3.Complete the square: y = (x - 3)^2 + 2. The squared term is never negative, so the smallest value of y is 2, reached at x = 3.

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7. 7Quadratic functionsChallenge

   The parabola y = -(x - 5)² + 12 has which of the following?The parabola y = -(x - 5)^2 + 12 has which of the following?

   • AA minimum of -12 at x = 5A minimum of -12 at x = 5
   • BA maximum of 5 at x = 12A maximum of 5 at x = 12
   • CA minimum of 12 at x = 5A minimum of 12 at x = 5
   • DA maximum of 12 at x = 5A maximum of 12 at x = 5
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   Answer: A maximum of 12 at x = 5A maximum of 12 at x = 5

   The negative sign in front of the squared term opens the parabola downward, so the vertex is a maximum. Vertex form gives the vertex at (5, 12).The negative sign in front of the squared term opens the parabola downward, so the vertex is a maximum. Vertex form gives the vertex at (5, 12).

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8. 8Quadratic functionsChallenge

   The graph of y = x² + kx + 16 touches the x-axis at exactly one point. What are the possible values of k?The graph of y = x^2 + kx + 16 touches the x-axis at exactly one point. What are the possible values of k?

   • A8 and -88 and -8
   • B16 and -1616 and -16
   • C4 and -44 and -4
   • D8 only8 only
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   Answer: 8 and -88 and -8

   One touching point means the discriminant is zero: k² - 64 = 0, so k = 8 or k = -8. Reporting only the positive value answers half the question.One touching point means the discriminant is zero: k^2 - 64 = 0, so k = 8 or k = -8. Reporting only the positive value answers half the question.

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9. 9Quadratic functionsChallenge

   What is the minimum value of f(x) = x² - 6x + 11?What is the minimum value of f(x) = x^2 - 6x + 11?

   • A22
   • B33
   • C1111
   • D-2-2
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   Answer: 22

   Completing the square gives f(x) = (x - 3)² + 2. A square is never negative, so the smallest value is 2, reached at x = 3. The choice 3 is where the minimum occurs rather than what it is, and 11 is the value at x = 0.Completing the square gives f(x) = (x - 3)^2 + 2. A square is never negative, so the smallest value is 2, reached at x = 3. The choice 3 is where the minimum occurs rather than what it is, and 11 is the value at x = 0.

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10. 10Quadratic functionsChallenge

    The graph of y = a(x - 2)² + k has vertex (2, 3) and passes through (0, 7). What is the value of a?The graph of y = a(x - 2)^2 + k has vertex (2, 3) and passes through (0, 7). What is the value of a?

    • A44
    • B22
    • C11
    • D141/4
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    Answer: 11

    The vertex gives k = 3. Substituting the point: 7 = a(0 - 2)² + 3 = 4a + 3, so 4a = 4 and a = 1. Forgetting to square the -2 gives a = 2.The vertex gives k = 3. Substituting the point: 7 = a(0 - 2)^2 + 3 = 4a + 3, so 4a = 4 and a = 1. Forgetting to square the -2 gives a = 2.

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