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-

x_{1} = 0, y_{1} = -2;

x_{2} = 1, y_{2} = -5;

x_{3} = 2, y_{3} = -14;

x_{4} = 3, y_{4} = -29;

x_{5} = 4, y_{5} = -50

a). Difference between 2 consecutive x values-

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

dx_{2} = x3 - 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} = -5 - (-2) = -5 + 2 = -3

dy_{2} = y_{3} - y_{2} = -14 - (-5) = -14 + 5 = -9

dy_{3} = y_{4} - y_{3} = -29 - (-14) = -29 + 14 = -15

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

dy_{2} - dy_{1} = -9 - (-3) = -9 + 3 = -6

dy_{3} - dy_{2} = -15 - (-9) = -15 + 9 = -6

dy_{4} - dy_{3} = -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.

_{1}= 0, y

_{1}= -2;

x

_{2}= 1, y

_{2}= -5;

x

_{3}= 2, y

_{3}= -14;

x

_{4}= 3, y

_{4}= -29;

x

_{5}= 4, y

_{5}= -50

a). Difference between 2 consecutive x values-

dx

_{1}= x

_{2}- x

_{1}= 1 - 0 = 1

dx

_{2}= x3 - 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}= -5 - (-2) = -5 + 2 = -3

dy

_{2}= y

_{3}- y

_{2}= -14 - (-5) = -14 + 5 = -9

dy

_{3}= y

_{4}- y

_{3}= -29 - (-14) = -29 + 14 = -15

dy

_{4}= y

_{5}- y

_{4}= -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-

dy

_{2}- dy

_{1}= -9 - (-3) = -9 + 3 = -6

dy

_{3}- dy

_{2}= -15 - (-9) = -15 + 9 = -6

dy

_{4}- dy

_{3}= -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.