Zeros

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Key Questions

  • The zeros of a function represent the x value(s) that result in the y value being 0.

    The zeros of a function represent the x-intercept(s) when the function is graphed.

    The zeros of a function represent the root(s) of a function.

    The zeros of a function represent the solution(s) of a function.

  • Answer:

    Graph the function on a graphing calculator to see what the x-coordinates are where the function intersects the x-axis.

    Explanation:

    The zeros of a function are found by determining what x-values will cause the y-value to be equal to zero. One way to find the zeros is to graph the function on a graphing calculator to see what the x-coordinates are where the function intersects the x-axis.

  • A zero of a function is an interception between the function itself and the X-axis.
    The possibilities are:

    • no zero (e.g. #y=x^2+1#) graph{x^2 +1 [-10, 10, -5, 5]}
    • one zero (e.g. #y=x#) graph{x [-10, 10, -5, 5]}
    • two or more zeros (e.g. #y=x^2-1#) graph{x^2-1 [-10, 10, -5, 5]}
    • infinite zeros (e.g. #y=sinx#) graph{sinx [-10, 10, -5, 5]}

    To find the eventual zeros of a function it is necessary to solve the equation system between the equation of the function and the equation of the X-axis (#y=0#).

  • Answer:

    If algebraic solutions are not usable, try Newton's method or similar to find numeric approximations.

    Explanation:

    Quintics and other more complicated functions

    If #f(x)# is a well behaved continuous, differentiable function - e.g. a polynomial, then you can find its zeros using Newton's method.

    Starting with an approximation #a_0#, iterate using the formula:

    #a_(i+1) = a_i - f(a_i)/(f'(a_i))#

    For example, if #f(x) = x^5+x+3#, then #f'(x) = 5x^4+1# and you would iterate using the formula:

    #a_(i+1) = a_i - (a_i^5+a_i+3)/(5a_i^4+1)#

    Putting this into a spreadsheet with #a_0 = -1#, I got the values:

    #a_0=-1#

    #a_1 = -1.166666666666667#

    #a_2 = -1.134701651504899#

    #a_3 = -1.133002126375077#

    #a_4 = -1.132997565917805#

    #a_5 = -1.132997565885065#

    #a_6 = -1.132997565885065#

    If #f(x)# has several Real zeros, then you may find them by choosing different values of #a_0#.

    Newton's method can also be used to find Complex zeros in a similar way, but you may prefer to use methods like Durand-Kerner to find all zeros at once.

Questions