Quadratic Functions and Their Graphs

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Graphing Basic Parabolic Functions.wmv

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Key Questions

  • A quadratic function has the general form:
    #y=ax^2+bx+c#

    (where #a,b and c# are real numbers) and is represented graphically by a curve called PARABOLA that has a shape of a downwards or upwards U.
    The main features of this curve are:
    1) Concavity: up or down. This depends upon the sign of the real number #a#:
    enter image source here
    2) Vertex. The vertex is the highes or lowest point of the parábola.
    the coordinates of this point are:
    #x=-b/(2a)# and #y=-Delta/(4a)#
    Where #Delta=b^2-4ac#

    3) point of intercept with the y axis. This is the point where the parábola crosses the y axis and has coordinates: #(0,c)#

    enter image source here

    4) Possible points of intercept with the x axis (there also can be none). These are the points where the parábola crosses the x axis.
    They are obtained by putting y=0 and solving for x the 2nd degree equation: #ax^2+bx+c=0#, which will give the x coordinates of these points (2 solutions)
    Depending on the discriminant #Delta=b^2-4ac# if it is <0 the parábola does not cross the x axis.
    enter image source here

  • There are two forms a quadratic function could be written in: standard or vertex form. Here are the following ways you can determine the vertex and direction dependent on the form:

    Standard Form (#f(x)=ax^2 + bx + c#)
    1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
    2. Vertex can be found by #x= -b/(2a)# and then plugging in that value to find y.
    Here is an example:
    #y = -2x^2 + 4x - 3#, Faces downward since a = -2.
    to find the vertex: #x= (-4)/(2(-2))=(-4)/ (-4) = 1#
    then plug that value into the equation#y = -2x^2 + 4x - 3#
    #y = -2(1)^2 + 4(1) - 3 #
    #y = -2(1) +4(1) - 3 #
    #y = -2 + 4 -3 #
    #y = -1 #
    Vertex is (1,-1)

    Vertex Form (#y=a(x-h)^2 +k#)
    1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
    2. Vertex is (h,k).
    Here is an example:
    #y = -3(x-2)+6# Faces down since a = -3 and the vertex is (2, 6).

  • A quadratic function is one of general form:
    #y=ax^2+bx+c#

    where a, b and c are real numbers.

    This function can be plotted giving a PARABOLA (a curve in the shape of an upward or downward U)

    To find the x intercepts you must put y=0; in this way you fix at zero the coordinate y of the points you are seeking.
    You are left with finding the coordinate x of the points.
    If y=0 you are left with: #0=ax^2+bx+c# which is a second degree equation.
    By solving this equation you'll find two values of x (x1 and x2) that together with y=0 will give you the intercepts:
    intercept 1: (x1 , 0)
    intercept 2: (x2 , 0)

    enter image source here

    Remember that a second degree equation can also have solutions:
    - coincident (the intercept is the VERTEX of the parabola)
    - imaginary (The parabola does not cross the x axis)
    Depending upon the discriminant of the equation.

  • The graphs of quadratic functions are called parabolas, and they look like the letter "U" right side up or up side down.


    I hope that this was helpful.

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