Question
What is the error in the student’s reasoning below ? Describe how to correct the statement ?
Hint:
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.
The correct answer is: The error in students reasoning given is that he/she did not consider the rate of change in x variables and the correct statement would be- "The data can not be modelled with a linear function because the x variables are not at a constant interval."
Step-by-step solution:-
From the above definition, it is clear that for a function to be classified as a linear function, the rate of change in its y variables has to be constant for a given constant change in its x variables.
From the given table, we can observe that-
x1 = -3, y1 = -8;
x2 = -1, y2 = -2;
x3 = 0, y3 = 4;
x4 = 1, y4 = 10 and
x5 = 3, y5 = 16
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = -1 - (-3) = -1 + 3 = 2
dx2 = x3 - x2 = 0 - (-1) = 0 + 1 = 1
dx3 = x4 - x3 = 1 - 0 = 1
dx4 = x5 - x4 = 3 - 1 = 2
b). Difference between 2 consecutive y values-
dy1 = y2 - y1 = -2 - (-8) = -2 + 8 = 6
dy2 = y3 - y2 = 4 - (-2) = 4 + 2 = 6
dy3 = y4 - y3 = 10 - 4 = 6
dy4 = y5 - y4 = 16 - 10 = 6
We can observe that although the rate of change in y variable seems to be constant @ 6, rate of change in x variables is not constant. i.e. some readings in x variable are missing that leads to a erroneous conclusion that first difference is constant.
Hence, The error in students reasoning given is that he/she did not consider the rate of change in x variables.
Also, the correct statement would be-
"The data can not be modelled with a linear function because the x variables are not at a constant interval."
Final Answer:-
∴ The error in students reasoning given is that he/she did not consider the rate of change in x variables and the correct statement would be- "The data can not be modelled with a linear function because the x variables are not at a constant interval."
x2 = -1, y2 = -2;
x3 = 0, y3 = 4;
x4 = 1, y4 = 10 and
x5 = 3, y5 = 16
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = -1 - (-3) = -1 + 3 = 2
dx2 = x3 - x2 = 0 - (-1) = 0 + 1 = 1
dx3 = x4 - x3 = 1 - 0 = 1
dx4 = x5 - x4 = 3 - 1 = 2
b). Difference between 2 consecutive y values-
dy1 = y2 - y1 = -2 - (-8) = -2 + 8 = 6
dy2 = y3 - y2 = 4 - (-2) = 4 + 2 = 6
dy3 = y4 - y3 = 10 - 4 = 6
dy4 = y5 - y4 = 16 - 10 = 6
We can observe that although the rate of change in y variable seems to be constant @ 6, rate of change in x variables is not constant. i.e. some readings in x variable are missing that leads to a erroneous conclusion that first difference is constant.
Hence, The error in students reasoning given is that he/she did not consider the rate of change in x variables.
Also, the correct statement would be-
"The data can not be modelled with a linear function because the x variables are not at a constant interval."
Final Answer:-
∴ The error in students reasoning given is that he/she did not consider the rate of change in x variables and the correct statement would be- "The data can not be modelled with a linear function because the x variables are not at a constant interval."
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