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22 June, 02:05

What are the key aspects of the graph of f (x) = x2 - 62,

where b is a real number?

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  1. 22 June, 02:17
    0
    The x intercepts are b, - b

    The y-intercept is - b²

    The vertex is (0, - b²)

    The axis of symmetry of the function is the y-axis

    Step-by-step explanation:

    * Lets explain the key aspects of the quadratic function

    - There are many key aspects in a quadratic graph such as:

    # x-intercepts

    # y-intercept

    # The vertex

    # Axis of symmetry

    * Lets solve the problem

    ∵ f (x) = x² - b² is a quadratic function

    ∵ The general form of the quadratic function is f (x) = Ax² + B (x) + C

    where A, B, C are constant

    ∴ A = 1, B = 0, C = - b²

    - To find the x-intercept put f (x) = 0

    ∵ f (x) = x² - b²

    ∴ x² - b² = 0 ⇒ add b² for both sides

    ∴ x² = b² ⇒ take √ for both sides

    ∴ x = ± b

    ∴ The x intercepts are b, - b

    - To find the y-intercept put x = 0

    ∵ f (x) = x² - b²

    ∴ f (0) = 0 - b²

    ∴ f (0) = - b²

    ∴ The y-intercept is - b²

    - The vertex of the function is (h, k) where h = - B/2A and k = f (h)

    ∵ A = 1 and B = 0

    ∴ h = 0/2 (1) = 0

    ∵ k = f (h)

    ∴ k = f (0) = 0 - b² = - b²

    ∴ k = - b²

    ∴ The vertex is (0, - b²)

    - The axis of symmetry of the quadratic function is a vertical line

    passes through the vertex of it and its equation is x = h

    ∵ h = 0

    ∴ The equation of the axis of symmetry is x = 0

    ∵ The equation of the y-axis is x = 0

    ∴ The axis of symmetry of the function is the y-axis
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