Question

# Function f represents the population , in millions , of Franklin x years from now. Function g represents the population , in millions , of Georgetown x years from now, If the pattern shown in the table continue , will franklin always have a greater population than Georgetown ? 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. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.

## The correct answer is: Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.

### Step-by-step solution:-

From the given table, we observe the following readings-

x_{1} = 0, y_{1}(f) = 3.4, y_{1}(g) = 2.4;

x_{2} = 1, y_{2}(f) = 5.6, y_{1}(g) = 3.6;

x_{3} = 2, y_{3}(f) = 7.8, y_{1}(g) = 5.4;

x_{4} = 3, y_{4}(f) = 10, y_{1}(g) = 8.1

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

a). For Franklin-

Difference between 2 consecutive y values-

dy_{1} = y_{2} - y_{1} = 5.6 - 3.4 = 2.2

dy_{2} = y_{3} - y_{2} = 7.8 - 5.6 = 2.2

dy_{3} = y_{4} - y_{3} = 10 - 7.8 = 2.2

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. 2.2.

Hence, the given function is a linear function.

b). For Georgetown-

Difference between 2 consecutive y values-

dy_{1} = y_{2} - y_{1} = 3.6 - 2.4 = 1.2

dy_{2} = y_{3} - y_{2} = 5.4 - 3.6 = 1.8

dy_{3} = y_{4} - y_{3} = 8.1 - 5.4 = 2.7

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.

Now, ratio between 2 consecutive y values-

= 3.6/2.4 = 1.5

= 5.4/3.6 = 1.5

= 8.1/5.4 = 1.5

We observe that difference between 2 consecutive y values is constant i.e. 1.5.

Hence, the given function is an exponential function.

We know from the above calculations that the population growth of Franklin is a Linear function and that of Georgetown is an exponential function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing at an exponential level. Hence, the population of Georgetown will far exceed that of Franklin in the long run.

Final Answer:-

∴ Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.

_{1}= 0, y

_{1}(f) = 3.4, y

_{1}(g) = 2.4;

x

_{2}= 1, y

_{2}(f) = 5.6, y

_{1}(g) = 3.6;

x

_{3}= 2, y

_{3}(f) = 7.8, y

_{1}(g) = 5.4;

x

_{4}= 3, y

_{4}(f) = 10, y

_{1}(g) = 8.1

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

a). For Franklin-

Difference between 2 consecutive y values-

dy

_{1}= y

_{2}- y

_{1}= 5.6 - 3.4 = 2.2

dy

_{2}= y

_{3}- y

_{2}= 7.8 - 5.6 = 2.2

dy

_{3}= y

_{4}- y

_{3}= 10 - 7.8 = 2.2

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. 2.2.

Hence, the given function is a linear function.

b). For Georgetown-

Difference between 2 consecutive y values-

dy

_{1}= y

_{2}- y

_{1}= 3.6 - 2.4 = 1.2

dy

_{2}= y

_{3}- y

_{2}= 5.4 - 3.6 = 1.8

dy

_{3}= y

_{4}- y

_{3}= 8.1 - 5.4 = 2.7

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.

Now, ratio between 2 consecutive y values-

= 3.6/2.4 = 1.5

= 5.4/3.6 = 1.5

= 8.1/5.4 = 1.5

We observe that difference between 2 consecutive y values is constant i.e. 1.5.

Hence, the given function is an exponential function.

We know from the above calculations that the population growth of Franklin is a Linear function and that of Georgetown is an exponential function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing at an exponential level. Hence, the population of Georgetown will far exceed that of Franklin in the long run.

Final Answer:-

∴ Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.