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Question

Determine whether a linear , quadratic or exponential function is the best model for the data in each table .

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. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.

The correct answer is: The given function is an exponential function.


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

    x1 = 0, y1 = 1;
    x2 = 1, y2 = 3;
    x3 = 2, y3 = 9;
    x4 = 3, y4 = 27;
    x5 = 4, y5 = 81
    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 = 3 - 1 = 2
                                                                                                   dy2 = y3 - y2 = 9 - 3 = 6
                                                                                                 dy3 = y4 - y3 = 27 - 9 = 18
                                                                                                dy4 = y5 - y4 = 81 - 27 = 54
    We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
    Hence, the given function is not a linear function.
    b). Now, difference between 2 consecutive differences for y values-
                                                                                                       dy2 - dy1 = 6 - 2 = 4
                                                                                                     dy3 - dy2 = 18 - 6 = 12
                                                                                                    dy4 - dy3 = 54 - 18 = 36
    We observe that the difference of differences of 2 consecutive y values are also not constant.
    Hence, the given function is not a quadratic function.
    c). Now, ratio between 2 consecutive y values-
                                                                                                          y2/y1 = 3/1 = 3
                                                                                                          y3/y2 = 9/3 = 3
                                                                                                         y4/y3 = 27/9 = 3
                                                                                                        y5/y4 = 81/27 = 3
    We observe that difference between 2 consecutive y values is constant i.e. 3.
    Hence, the given function is an exponential function.
    Final Answer:-
    ∴ The given function is an exponential function.

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