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    Graphing Functions: Completing the Square, Quadratic Functions, Axis of Symmetry, and Intercepts

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    1. Using completing the square to describe the graph of the following function. Support your answer graphically.
    f(x) = -2x^2 + 4x + 1
    2. Graph the function: g(x) = (x-2)^3
    3. Determine the quadratic function f whose vertex is (3, -2) and passes through (2, 1)
    4. Graph the line containing the point P and having slope m: P = (2, -7); m = 0
    5. Graph the function and state the vertex, the axis of symmetry, the intercepts, if any: f(x) = x^2-6x+5
    What is the vertex?
    What is the axis of symmetry?
    What are the intercepts, if any?

    © BrainMass Inc. brainmass.com December 24, 2021, 11:09 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/completing-square-quadratic-functions-axis-symmetry-543240

    SOLUTION This solution is FREE courtesy of BrainMass!

    1.

    This parabola has a vertex at

    We see that as the function approaches , so the vertex point is a maxima (the parabola opens down).
    The parabola intersects the x-axis when


    Which gives:

    It intersects the y-axis when x=0 and this gives:

    To summarize:
    The parabola has a vertex which is a maxima at so its symmetry axis is x=1
    It intersects the x-axis at and
    It intersects the y-axis at

    It looks like:


    2.
    The function is

    This is a third degree polynomial that has one root with multiplicity 3.
    When we have
    The graph is the same as that of only transposed 2 units to the right.
    It intersects the y-axis when x=0 so the intersection point is
    It looks like:


    3.
    The general form of a quadratic function is:

    Where is the vertex.
    In our case the vertex is , hence our parabola has the form:

    We require that , and this gives for a:

    Therefore:


    Opening the parenthesis

    It looks like:


    4.
    A line with slope m and y-intercept b has the canonical equation:

    In our case m=0, so the line's equation is:

    This is an equation of a line parallel to the x axis (y is constant)
    Since the line passes through the point (2,-7) we see this constant must be -7, hence the line equation is

    And it looks like:

    5.
    The function is:

    We convert it to the canonical form :

    We see that the vertex is

    And it is a minima (the parabola opens upward)
    The function is symmetric about the vertical axis that passes through the vertex:

    It intersects the x axis when which gives:

    The x-intercepts are and
    It intersects the y-axis when x=0 which gives:

    The y-intercept is

    It looks like:

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:09 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/completing-square-quadratic-functions-axis-symmetry-543240

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