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Math Help - Characteristics of the graph

  1. #1
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    Characteristics of the graph

    Discuss the characteristics of the graph that arise when a function has a factor the appears twice, three times, four times, etc...

    No idea what I'm suppose to do any help?
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  2. #2
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    Hello lost in functions

    Welcome to Math Help Forum!
    Quote Originally Posted by lost in functions View Post
    Discuss the characteristics of the graph that arise when a function has a factor the appears twice, three times, four times, etc...

    No idea what I'm suppose to do any help?
    If a function f(x) has a factor (x-a) then the equation f(x) = 0 has a root (a solution) at x = a. So in this case, when x = a, the graph y = f(x) is 'at' the x-axis, because the x-axis is where y = 0. OK so far?

    Now I say 'at' the x-axis because various things might happen:

    • The graph may cut the x-axis, starting on one side of the axis, crossing it at x = a and emerging on the other side. This will happen if the factor (x-a) appears just once in f(x).


    • The graph may be a tangent to the x-axis where x = a. So the graph comes up to the x-axis, touches it and then returns on the same side of the axis from whence it came. This corresponds to (x-a)^2 being a factor of f(x). A tangent has what's called '2-point contact' with the line or curve that it touches.


    • The graph may do both of the above! It may be a tangent at x = a (in other words it is horizontal at this point) but it may also cross the axis and emerge on the other side. This is what happens if (x-a)^3 is a factor of f(x). This is a point of inflexion, and is sometimes referred to as 3-point contact.


    • If (x-a)^4 is a factor then we have 4-point contact, and the curve touches the x-axis and returns on the side from whence it came.


    • If (x-a)^5 is a factor we have 5-point contact and the graph crosses the axis.

    ... and so on.

    Grandad
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