Quadratic Functions and Transformations
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Questions and Answers

For the quadratic function $f(x) = a(x + 1)(x + 5)$, what are the x-intercepts of the graph?

  • (1, 0) and (5, 0)
  • (-1, 0) and (-5, 0) (correct)
  • (0, 1) and (0, 5)
  • (0, -1) and (0, -5)

    • The equation $kx^2 + 4kx - 1 = k$ always has two real, distinct roots for any real number k.

    False (B)

    If the y-intercept of the function $f(x) = a(x + 1)(x + 5)$ is (0, -10), what is the value of a?

    2

    The axis of symmetry for the quadratic $f(x) = a(x + 1)(x + 5)$ is $x = $ ______.

    <p>-3</p> Signup and view all the answers

    Match each part of the quadratic function $f(x) = 16 - x^2$ to its description:

    <p>x-intercepts = Points where the graph intersects the x-axis y-intercept = Point where the graph intersects the y-axis Maximum value = The highest point on the graph</p> Signup and view all the answers

    Given $f(x) = log_k(8x - 2x^2)$, for what value of $k$ does the equation $f(x) = 3$ have exactly one solution?

    <p>4 (D)</p> Signup and view all the answers

    If the graphs of $f(x) = 16 - x^2$ and $g(x) = (x - 4)^2 + k$ intersect only once, then k = 0.

    <p>False (B)</p> Signup and view all the answers

    For the rectangle ABCD, where A and B are on the x-axis and C and D are on the graph of $f(x) = 16 - x^2$, if $OA = a$, what is an expression for the area of rectangle ABCD?

    <p>$32a - 2a^3$</p> Signup and view all the answers

    Flashcards

    Quadratic Function

    A polynomial function of the form f(x) = ax² + bx + c.

    X-Intercepts

    Points where the graph crosses the x-axis (y=0).

    Y-Intercept

    Point where the graph crosses the y-axis (x=0).

    Axis of Symmetry

    Vertical line that divides the parabola into two mirrored halves.

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    Vertex of a Parabola

    The highest or lowest point on the graph.

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    Reflection in Y-Axis

    Transformation that flips the graph over the y-axis.

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    Real Distinct Roots

    Two different solutions to a quadratic equation.

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    Area of Rectangle related to Quadratic

    Area calculated as a function of the width (a) and a quadratic expression.

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    Study Notes

    Question 26

    • A quadratic function is given as f(x) = a(x+1)(x+5) for x ∈ R and a ∈ Z.
    • The graph of f has x-intercepts at (p, 0) and (q, 0) and a y-intercept at (0, -10).
    • p = -1, q = -5.
    • a = 2.
    • The equation of the axis of symmetry is x = -3.
    • The coordinates of the vertex are (-3, -16).
    • A transformation is applied to the graph of f, involving a reflection in the y-axis and a translation.
    • The new coordinates of the point Q mapped from P(-2, 6) are (2, 6).

    Question 27

    • The quadratic equation is kx² + 4kx – 10 = 0
    • For real and distinct roots, the discriminant must be positive, Δ > 0.
    • Possible values for k are k < -5 or k > 2.
    • Given k = 3 the roots are in the form q/√3 where p, q ∈ Z
    • The roots are 2 ± √2.

    Question 28

    • f(x) = logk(8x – 2x²), for 0 < x < 4, and k > 0.
    • The equation f(x) = 3 has exactly one solution.
    • k = 2

    Question 29

    • f(x) = x² - kx + 5, for x ∈ R
    • g(x) = x + 4, for x ∈ R.
    • The range of f(x) is given by f(x) ≥ c. where c is the minimum value of the parabola f(x).
    • Given that (gof)(x) ≤ 0 for all real values of x, the set of possible values for k is k ≥ 4.

    Question 30

    • f(x) = 16 – x²
    • x-intercepts are (4, 0) and (-4, 0).
    • The area of rectangle ABCD is given by 32a – 2a³.
    • The value of a that maximizes the area is 4/√3.

    Question 31

    • f(x) = 2(x - h₁)² + k₁
    • g(x) = (x - h₂)² + k₂
    • The vertex of f is (m, -m²) and the vertex of g is (-m, -m).
    • 0 < m < 1
    • The graphs intersect at exactly one point.
    • m = 1/√3.

    Question 32

    • f(x) = p(x - 1)(x-q).
    • y-intercept of f(x) is (0, -3)
    • Axis of symmetry is x = 2.
    • q = 3.
    • p = 1.
    • Possible values for k are k = 2, or k = -2/3 ± √13/3

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    IB Math AA SL Past Paper Questions - Quadratics PDF

    Description

    This quiz covers various aspects of quadratic functions including their transformations, the properties of their graphs, and the implications of their coefficients. It delves into the determination of roots, the effects of parameters, and the analysis of function compositions. Perfect for students looking to solidify their understanding of quadratic equations and functions.

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