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    Maths-
    General
    Easy

    Question

    Does a linear , quadratic , or exponential function best model the data ? Explain.

    hintHint:

    1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.
    2. When the difference between 2 consecutive differences for output values (y values) for a given constant change in the input values (x values) is constant. i.e. dy(n)- dy(n-1) is constant for any value of n, the function is known as a quadratic function.
    3. When the ratio between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant i.e. fraction numerator y left parenthesis n right parenthesis over denominator y left parenthesis n minus 1 right parenthesis end fraction is constant for any value of n, the function is known as an exponential function.

    The correct answer is: The given table can be modelled as an exponential function because the ratio between any 2 consecutive output values i.e. y values is constant i.e. 2.

      Step-by-step solution:-
      From the given table, we observe the following readings-

      x1 = 0, y1 = -2;
      x2 = 1, y2 = -5;
      x3 = 2, y3 = -14;
      x4 = 3, y4 = -29;
      x5 = 4, y5 = -50
      a). Difference between 2 consecutive x values-
                                                                                              dx1 = x2 - x1 = 1 - 0 = 1
                                                                                              dx2 = x3 - x2 = 2 - 1 = 1
                                                                                              dx3 = x4 - x3 = 3 - 2 = 1
                                                                                              dx4 = x5 - x4 = 4 - 3 = 1
      Difference between 2 consecutive y values-
                                                                                   dy1 = y2 - y1 = -5 - (-2) = -5 + 2 = -3
                                                                                 dy2 = y3 - y2 = -14 - (-5) = -14 + 5 = -9
                                                                              dy3 = y4 - y3 = -29 - (-14) = -29 + 14 = -15
                                                                              dy4 = y5 - y4 = -50 - (-29) = -50 + 29 = -21
      We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference are not constant.
      Hence, the given function is not a linear function.
      b). Now, difference between 2 consecutive differences for y values-
                                                                                      dy2 - dy1 = -9 - (-3) = -9 + 3 = -6
                                                                                    dy3 - dy2 = -15 - (-9) = -15 + 9 = -6
                                                                                  dy4 - dy3 = -21 - (-15) = -21 + 15 = -6
      We observe that the difference of differences of 2 consecutive y values are constant i.e. -6.
      Hence, the given function is a quadratic function.
      Final Answer:-
      ∴ The given function is a quadratic function.

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