Question
Does a linear , quadratic , or exponential function best model the data ? Explain.
Hint:
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. 
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 = -2, y1 = 4;
x2 = -1, y2 = 12;
x3 = 0, y3 = 36;
x4 = 1, y4 = 108;
x5 = 2, y5 = 324.
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = -1 - (-2) = -1 + 2 = 1
dx2 = x3 - x2 = 0 - (-1) = 0 + 1 = 1
dx3 = x4 - x3 = 1 - 0 = 1
dx4 = x5 - x4 = 2 - 1 = 1
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 12 - 4 = 8
dy2 = y3 - y2 = 36 - 12 = 24
dy3 = y4 - y3 = 108 - 36 = 72
dy4 = y5 - y4 = 324 - 108 = 216
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 = 24 - 8 = 16
dy3 - dy2 = 72 - 24 = 48
dy4 - dy3 = 216 - 72 = 144
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-
= 
= 3
= 
= 3
= 
= 3
= 
= 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.
x2 = -1, y2 = 12;
x3 = 0, y3 = 36;
x4 = 1, y4 = 108;
x5 = 2, y5 = 324.
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = -1 - (-2) = -1 + 2 = 1
dx2 = x3 - x2 = 0 - (-1) = 0 + 1 = 1
dx3 = x4 - x3 = 1 - 0 = 1
dx4 = x5 - x4 = 2 - 1 = 1
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 12 - 4 = 8
dy2 = y3 - y2 = 36 - 12 = 24
dy3 = y4 - y3 = 108 - 36 = 72
dy4 = y5 - y4 = 324 - 108 = 216
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 = 24 - 8 = 16
dy3 - dy2 = 72 - 24 = 48
dy4 - dy3 = 216 - 72 = 144
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-
= = 3 = = 3 = = 3 = = 3We 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.
Check for extraneous solutions
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