Question
Determine whether a linear , quadratic or exponential function is the best model for the data in each table .
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. 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.
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.
,where H is the height in centimeters and M is the mass in kilograms. A doctor calculates a particular dose of medicine for a patient whose BSA is less than 1.9. If the patient is 160 cm tall, what must the mass of the person be for the dose to be appropriate?
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