Question

07.03, 07.05 HC)

Use the function f(x) = −16×2 + 22x + 3 to answer the questions.

Part A: Completely factor f(x). (2 points)

Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)

Part C: Describe the end behavior of the graph of f(x). Explain. (2 points)

Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to dra

Answers

    1. The x-intercepts of the graph of f(x) are (-1/8, 3/2).
    2. The parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.
    3. Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
    4. Determine the y-intercept (3).
    5. Connect all the known points on the graph to form a downward parabola.

    What is a quadratic function?

    A quadratic function can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.
    In Mathematics, the graph of any quadratic function is parabolic because it is a u-shaped curve. For the given quadratic function, the graph is a downward parabola because the coefficient of x² is negative.

    How to determine the x-intercepts?

    In order to determine the x-intercepts of the graph of f(x), we would make to be y = 0.
    By rearranging and completely factoring the given quadratic function, we have:
    f(x) = -16x² + 22x + 3
    f(x) = 16x² – 22x – 3
    16x² – 24x + 2x – 3 = 0
    8x(2x – 3) + 1(2x – 3) = 0
    (8x + 1)(2x – 3) = 0
    x₁ ⇒ 8x = -1      ⇒     x₁ = -1/8.
    x₂ ⇒ 2x = 3      ⇒     x₂ = 3/2.    
    Therefore, the x-intercepts of the graph of f(x) are (-1/8, 3/2).

    How to describe the end behavior?

    Since the parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.
    V(x) = -b/2a
    V(x) = -22/2(-16)
    V(x) = -22/-32
    V(x) = 11/16
    V(y) = -D/4a
    V(y) = -16(11/16)² + 22(11/16) + 3
    V(y) = -16(121/256) + 121/8 + 3
    V(y) = -121/16 + 121/8 + 3
    V(y) = (-121 + 242 + 48)/16
    V(y) = 169/16.

    What are the steps you would use to graph f(x)?

    • Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
    • Determine the y-intercept (3).
    • Connect all the known points on the graph to form a downward parabola.
    Read more on quadratic functions here: https://brainly.com/question/24020644
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