Question

# Ella wrote three different computer apps to analyze some data. The table show the time in millisecond y for each app to analyze data as a function of the number of data items x.

a. Use regression on a graphing calculator to find a function that models each data set . Explain your choice of model .

b. Make a conjecture about which app will require the most time as the number of data items gets very large. How could you support your conjecture

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: We should choose App B as it takes least time to process increased number of data items in the long run because it is a Linear function. Also, App A will take the most time as the number of data items get very large because it is an exponential function.

### Step-by-step solution:-

For App A-

Ratio -= = 3

= = 3

= = 3

= = 3

Since the ratio between all consecutive y values is constant, App A represents an Exponential function.

For App B-

First difference-

d_{1} = y_{2} - y_{1} = 5,040 - 4,042 = 998

d_{2} = y_{3} - y_{2} = 6,038 - 5,040 = 998

d_{3} = y_{4} - y_{3} = 7,036 - 6,038 = 998

d_{4} = y_{5} - y_{4} = 8,034 - 7,036 = 998

Since the first difference between all consecutive y values is constant i.e. 998, App B represents a Linear function.

For App C-

First difference-

d_{1} = y_{2} - y_{1} = 5,375 - 4,400 = 975

d_{2} = y_{3} - y_{2} = 6,550 - 5,375 = 1,175

d_{3} = y_{4} - y_{3} = 7,925 - 6,550 = 1,375

d_{4} = y_{5} - y_{4} = 9,500 - 7,925 = 1,575

Second difference-

d_{2} - d_{1} = 1,175 - 975 = 200

d_{3} - d_{2} = 1,375 - 1,175 = 200

d_{4} - d_{3} = 1,575 - 1,375 = 200

Since the second difference between all consecutive y values is constant i.e. 200, App C represents a Quadratic function.

Using regression on a graphing calculator, we find the functions that model data for App A, B & C as follows-

For App A- h(x) = 3x

For App B- f(x) = 50 + 998x

For App C- g(x) = 100x_{2} + 75x + 2,500

a). Since in the given data, y represents the time taken by app to analyze a certain quantity of data represented by x, we should choose the app that will take lesser time to analyze data as x increases.

Since App A and App C are represented by Exponential and Quadratic functions, respectively, these apps will take increasing amount of time to process as data quantity increases. Whereas, App B is represented by a Linear function which means that time taken by App B will increase at a constant rate as the data quantity increase.

Hence, the rate of increase in time taken by each app is the least for App B.

Hence, we should choose App B.

b). App A will require the most time as the number of data items gets very large because it is represented by an exponential function and we know that for an exponential function the rate of increase in y values for a given change in x values is exponential.

Final Answer:-

∴ We should choose App B as it takes least time to process increased number of data items in the long run because it is a Linear function.

Also, App A will take the most time as the number of data items get very large because it is an exponential function.

Ratio -= = 3

= = 3

= = 3

= = 3

Since the ratio between all consecutive y values is constant, App A represents an Exponential function.

For App B-

First difference-

d

_{1}= y

_{2}- y

_{1}= 5,040 - 4,042 = 998

d

_{2}= y

_{3}- y

_{2}= 6,038 - 5,040 = 998

d

_{3}= y

_{4}- y

_{3}= 7,036 - 6,038 = 998

d

_{4}= y

_{5}- y

_{4}= 8,034 - 7,036 = 998

Since the first difference between all consecutive y values is constant i.e. 998, App B represents a Linear function.

For App C-

First difference-

d

_{1}= y

_{2}- y

_{1}= 5,375 - 4,400 = 975

d

_{2}= y

_{3}- y

_{2}= 6,550 - 5,375 = 1,175

d

_{3}= y

_{4}- y

_{3}= 7,925 - 6,550 = 1,375

d

_{4}= y

_{5}- y

_{4}= 9,500 - 7,925 = 1,575

Second difference-

d

_{2}- d

_{1}= 1,175 - 975 = 200

d

_{3}- d

_{2}= 1,375 - 1,175 = 200

d

_{4}- d

_{3}= 1,575 - 1,375 = 200

Since the second difference between all consecutive y values is constant i.e. 200, App C represents a Quadratic function.

Using regression on a graphing calculator, we find the functions that model data for App A, B & C as follows-

For App A- h(x) = 3x

For App B- f(x) = 50 + 998x

For App C- g(x) = 100x

_{2}+ 75x + 2,500

a). Since in the given data, y represents the time taken by app to analyze a certain quantity of data represented by x, we should choose the app that will take lesser time to analyze data as x increases.

Since App A and App C are represented by Exponential and Quadratic functions, respectively, these apps will take increasing amount of time to process as data quantity increases. Whereas, App B is represented by a Linear function which means that time taken by App B will increase at a constant rate as the data quantity increase.

Hence, the rate of increase in time taken by each app is the least for App B.

Hence, we should choose App B.

b). App A will require the most time as the number of data items gets very large because it is represented by an exponential function and we know that for an exponential function the rate of increase in y values for a given change in x values is exponential.

Final Answer:-

∴ We should choose App B as it takes least time to process increased number of data items in the long run because it is a Linear function.

Also, App A will take the most time as the number of data items get very large because it is an exponential function.