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 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.
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.
About how far is the cable from roadway when you are 200 ft from middle of the bridge
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