Question

# Determine whether the data are best modelled by a linear , quadratic or exponential function ?

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 a linear function.

### Step-by-step solution:-

From the given table, we observe the following readings-

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

x_{2} = -1, y_{2} = 7;

x_{3} = 0, y_{3} = 12;

x_{4} = 1, y_{4} = 17;

x_{5} = 2, y_{5} = 22

a). Difference between 2 consecutive x values-

dx_{1} = x_{2} - x_{1} = -1 - (-2) = -1 + 2 = 1

dx_{2} = x_{3} - x_{2} = 0 - (-1) = 0 + 1 = 1

dx_{3} = x_{4} - x_{3} = 1 - 0 = 1

dx_{4} = x_{5} - x_{4} = 2 - 1 = 1

Difference between 2 consecutive y values-

dy_{1} = y_{2} - y1 = 7 - 2 = 5

dy_{2} = y_{3} - y_{2} = 12 - 7 = 5

dy_{3} = y_{4} - y_{3} = 17 - 12 = 5

dy_{4} = y_{5} - y_{4} = 22 - 17 = 5

We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. 5.

Hence, the given function is a linear function.

Final Answer:-

∴ The given function is a linear function.

_{1}= -2, y

_{1}= 2;

x

_{2}= -1, y

_{2}= 7;

x

_{3}= 0, y

_{3}= 12;

x

_{4}= 1, y

_{4}= 17;

x

_{5}= 2, y

_{5}= 22

a). Difference between 2 consecutive x values-

dx

_{1}= x

_{2}- x

_{1}= -1 - (-2) = -1 + 2 = 1

dx

_{2}= x

_{3}- x

_{2}= 0 - (-1) = 0 + 1 = 1

dx

_{3}= x

_{4}- x

_{3}= 1 - 0 = 1

dx

_{4}= x

_{5}- x

_{4}= 2 - 1 = 1

Difference between 2 consecutive y values-

dy

_{1}= y

_{2}- y1 = 7 - 2 = 5

dy

_{2}= y

_{3}- y

_{2}= 12 - 7 = 5

dy

_{3}= y

_{4}- y

_{3}= 17 - 12 = 5

dy

_{4}= y

_{5}- y

_{4}= 22 - 17 = 5

We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. 5.

Hence, the given function is a linear function.

Final Answer:-

∴ The given function is a linear function.