Question

# Determine whether a linear , quadratic or exponential function best models the data . Then use regression to find the function that models the data ?

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. Regression is a statistical tool used to find a model that can represent the relation between a given change in dependant variable (output values/ y values) for a given change in independent variable (input values/ x values).

Linear Equation using regression can be represented as-

Y = a + bX, where-

a =

b=

## The correct answer is: The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 100.02 - 10.55X.

### Step-by-step solution:-

From the given table, we observe the following readings-

x_{1} = 0, y_{1} = 100;

x_{2} = 1, y_{2} = 89.5;

x_{3} = 2, y_{3} = 78.9;

x_{4} = 3, y_{4} = 68.4;

x_{5} = 4, y_{5} = 57.8

a). Difference between 2 consecutive x values-

dx_{1} = x_{2} - x_{1} = 1 - 0 = 1

dx_{2} = x_{3} - x_{2} = 2 - 1 = 1

dx_{3} = x_{4} - x_{3} = 3 - 2 = 1

dx_{4} = x_{5} - x_{4} = 4 - 3 = 1

Difference between 2 consecutive y values-

dy_{1} = y_{2} - y_{1} = 89.5 - 100 = -10.5

dy_{2} = y_{3} - y_{2} = 78.9 - 89.5 = -10.6

dy_{3} = y_{4} - y_{3} = 68.4 - 78.9 = -10.5

dy_{4} = y_{5} - y_{4} = 57.8 - 68.4 = -10.6

We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is almost constant i.e. -10.5.

Hence, the given function is a linear function.

Using Linear Regression formula-

Y = a + bX, where-

a =

∴ a = [.......................... (As per adjacent table)

∴ a =

∴ a =

∴ a = 100.02

b =

∴ b =

∴ b =

∴ b =

∴ b = -10.55

∴ The Linear Equation is-

Y = a + bX

∴ Y = 100.02 + (-10.55)X

∴ Y = 100.02 - 10.55X

Final Answer:-

∴ The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 100.02 - 10.55X.

_{1}= 0, y

_{1}= 100;

x

_{2}= 1, y

_{2}= 89.5;

x

_{3}= 2, y

_{3}= 78.9;

x

_{4}= 3, y

_{4}= 68.4;

x

_{5}= 4, y

_{5}= 57.8

a). Difference between 2 consecutive x values-

dx

_{1}= x

_{2}- x

_{1}= 1 - 0 = 1

dx

_{2}= x

_{3}- x

_{2}= 2 - 1 = 1

dx

_{3}= x

_{4}- x

_{3}= 3 - 2 = 1

dx

_{4}= x

_{5}- x

_{4}= 4 - 3 = 1

Difference between 2 consecutive y values-

dy

_{1}= y

_{2}- y

_{1}= 89.5 - 100 = -10.5

dy

_{2}= y

_{3}- y

_{2}= 78.9 - 89.5 = -10.6

dy

_{3}= y

_{4}- y

_{3}= 68.4 - 78.9 = -10.5

dy

_{4}= y

_{5}- y

_{4}= 57.8 - 68.4 = -10.6

We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is almost constant i.e. -10.5.

Hence, the given function is a linear function.

Using Linear Regression formula-

Y = a + bX, where-

a =

∴ a = [.......................... (As per adjacent table)

∴ a =

∴ a =

∴ a = 100.02

b =

∴ b =

∴ b =

∴ b =

∴ b = -10.55

∴ The Linear Equation is-

Y = a + bX

∴ Y = 100.02 + (-10.55)X

∴ Y = 100.02 - 10.55X

Final Answer:-

∴ The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 100.02 - 10.55X.