C197- Applied Algebra
A job report displays monthly costs of advertising and monthly revenue. It shows that revenue is usually three times the cost of advertising. Assuming this pattern continues, what is the expected revenue, if the company spends $6,000 on advertising?
$18,000
In the function C(x) = 21.25 + 0.25x, C(x) represents the total cost, in dollars, for a passenger to ride a shuttle service for x miles. What is the total cost for a passenger to ride the shuttle for 50 miles?
$33.75
A cell phone company charges a fee of $90 per month of service. A customer's cost, C, for service can be represented by the function 𝐶(𝑚) = 90𝑚, where m represents the number of months. Find the total cost for six months of service.
$540
Larry is selling two different brands of computers: the Robin and the Falcon. The Robin is an older model that sells for $250 each. The Falcon is a newer model that sells for $600 each. The function F(x,y)=250x+600y𝐹(𝑥,𝑦)=250𝑥+600𝑦 models the total sales, in dollars, where x is the number of Robin computers and y is the number of Falcon computers sold. Evaluate F(12,7)𝐹(12,7) and interpret its meaning in the context.
$7,200
A company that sells running shoes has a yearly fixed cost of $500,000. Additionally, it costs the company $20 to produce each pair of shoes. The company's yearly cost is represented by the function 𝐶(𝑥) = 500,000 + 20𝑥, where x equals the number of shoes. If the company produces 10,000 pairs of shoes in a year, what is the yearly cost?
$700,000
Cheetah Computer's net profit, in millions of dollars, can be modeled by the function P(t)=2t−8𝑃(𝑡)=2𝑡−8, where t is the number of years since 2010. Which of the following coordinate points offer valid input and output pairs of the function?
(1, -6) and (5, 2)
For a school fundraiser, a local business will donate $1 for every $10 donated by the community. What would the corresponding coordinate be if the community raised $1,450?
(1450, 145)
You want to drain your rectangular in-ground swimming pool, which contains 9,000 gallons of water. The amount of water remaining in the pool can be represented by the function 𝑊(𝑡) = 9 - 2𝑡, where t is the time, in hours, after the draining began. The function, W(t), is measured in thousands of gallons. How many gallons of water are in the pool two hours after the draining procedure begins?
5,000 gallons
Suppose a short-distance towing company uses the function P(m) = 1.8m + 32 to determine the price, P, of towing a car m miles. What would the price be for the coordinate (10, P)?
50
A forensic pathologist wants to model a person's height, in inches, as a function of their femur length, in centimeters. This can be represented by the height function H(𝑓) = 0.91𝑓 + 24, where f is the length of the person's femur. If the femur length of a woman is 41 cm, what is her height?
61.31 inches
The distance in miles that a car drives can be represented by the function 𝐷(𝑔) = 28𝑔, where g is the number of gallons of gasoline. If you drive 224 miles, how much gas do you use?
8 gallons
Given the function 𝐹(𝑥) = −3𝑥 + 15, what is the value of 𝐹(2)?
9
4th Degree Polynomial:
A Quartic polynomial
Inverse function:
A function in which two variables are inversely proportional. Use division to solve instead of multiplication.
Which data representation method—data table or graph—would be most useful if you were hoping to examine trends in the data over time and why?
A graph would be better suited to this purpose than a data table, as a graph allows you to see aggregate trends in data without being bogged down by the values of individual data points.
Which regression model should you choose if a set of data's y-value increases sharply at first, and then becomes flat and approaches 10? Why?
A logistic model fits this set of data, which has a limit and is shaped like part of the letter S.
You want to rent a car for the weekend, so you compare prices for two different companies. Abbie's Rental Company charges a base rate of $100 to rent the car, with and additional price of $0.01 per mile. Best Rental charges a base rate of $50 to rent the car, with an additional price of $0.10 per mile. Which of the following correctly represents the functions of both companies' prices, where m represents the number of miles the rental car is driven?
Abbie's Rental Company: A(m)=0.01m+100𝐴(𝑚)=0.01𝑚+100 Best Rental: B(m)=0.10m+50
A new car loses 8% of its value every year. Which of these types of functions should be used in a regression to model the car's value?
An exponential model
Which regression model should you choose if a set of data's y-value decreases by a third, every five years? Why?
An exponential model should be chosen because the data has a constant ratio.
Michelle uses the function H(c)=0.02c𝐻(𝑐)=0.02𝑐 to make predictions for the number of hours her claims processors need to work in a week. In the function, H stands for the projected number of hours the processors work to process a particular number of claims, c. If Michelle has 2,500 health insurance claims to process for next week, what conclusion can she make about any overtime (working over 40 hours a week) her processors may be called on to do?
Based on the projections, Michelle's processors will each need to work about 10 hours of overtime. 𝐻(2500) = 0.02(2500)=50.
A local coffee shop found that its number of customers, N, can be predicted by the time of day in hours, t. The opening time of 5:00 a.m. corresponds to t = 0. How should the graph be interpreted between t = 5.5 and t = 9?
Between 10:30 a.m. and 2:00 p.m., the number of customers is increasing faster and faster and then increasing slower and slower.
The use of Brackets and parenthesis:
Brackets indicate that an endpoint is included, parenthesis are used when excluding an endpoint. Example: [92 , 200)
A company pays its employees a base salary each month plus a commission, C, based on the number of sales, s, made by an employee during the month. An employee made 8 sales in the last month and received a commission of $1,600. How would this be represented in function notation?
C(8) = 1600
Yara sells two brands of computers, the Flash and the Cheetah, at a trade show. Yara determines that the sale of one Flash computer equates to the sales of two Cheetah computers. Write a function describing the number of Cheetah sales, C, as a function of the number of Flash sales, f.
C(f)=2f
You own a blog. You earn 12% commission on sales whenever a visitor clicks on an ad and purchases something. Let C(s)𝐶(𝑠) represent the amount you earn from commission on sales, and let s represent your total sales. Which function represents the amount you earn from the sales commission?
C(s) = 0.12s
Many medical studies have shown a relationship between fast-food consumption and an increase in cholesterol. Which variable is the dependent variable?
Cholesterol increase
3rd Degree Polynomial:
Cubic Polynomial that has an S shape. I can have one or two inflection points and cross the x-axis up to three times.
You are reading a report that includes this notation: [1996, 2016]. What meaning should you understand from that notation?
Data between 1996 and 2016 is included, inclusive of both 1996 and 2016.
The number of gym memberships since January 1, 2015, for a particular gym is given by the function N(x) = 100 + 11x, where N is the number of gym members x months after January 1, 2015. How can the slope be interpreted?
Each month, 11 members join the gym.
Which one of these events allows you to predict the output based on an input?
Entering numbers on the TV remote
A rental car company charges a flat fee of $39 for a car rental. They charge an additional $0.62 for each mile driven. Which function F represents the cost to rent a car, where m represents the number of miles?
F(m) = 39 + 0.62m
You just started a lawn company. You spent $200 in initial costs, and you earn $150 per month. Which function can be used to predict your profit after x months?
F(x) =150x - 200
Larry is selling two different brands of computers: the Robin and the Falcon. The Robin is an older model that sells for $250 each. The Falcon is a newer model that sells for $600 each. Which function correctly models Larry's total sales for both computer types in this transaction?
F(x,y) = 250x + 600y, where x represents the number of Robin computers sold and y represents the number of Falcon computers sold.
A bank begins a marketing campaign to increase its number of customers. Each month, all new customers are put into a drawing. One customer then receives a $50 credit in a checking account. The amount of money given away in the drawing, G, is a function of the number of months, M, the drawing has been going on for. What is the correct function notation to represent the amount of money given away after the drawing has been going on for 5 months?
G(5) = 250
Yara sells two brands of computers, the Flash and the Cheetah, at a trade show. There are 100 Cheetah computers available. The offer includes discounted prices and a strict limitation on the number of products any one person can purchase. As a result, sales of one brand affects sales of the other. Yara determines that the sale of one Flash computer equates to two fewer sales of Cheetah computers. The function 𝐶(𝑓)= − 2𝑓 + 100 represents the number of Cheetah computers sold, where f is the number of Flash computers sold. Which of the following is a correct interpretation of that function?
If Yara sells 4 Flashes, she would sell 92 Cheetah computers.
Zini has a lemonade stand and she wants to know how many cups of lemonade she should sell to make a profit, represented by R(5) = 2.50 What does the above equation mean?
If Zini sells 5 cups of lemonade, the revenue is $2.50.
Which statement is true?
If a regression done with nine data points has a coefficient of determination of 0.95, the regression function can be used to predict data.
A server at a restaurant has realized that tips, T, in dollars are related to the number of hours, h, that have been worked on any given day. Tips can be modeled by the function T(h) = X. How can this function be interpreted?
If the server works h hours, the tips will be $X.
A local business is looking to become a franchise. The company wants to predict the number of franchises, F, given a certain number of years, Y. This prediction is modeled by the function F(Y). What can be concluded by F(5) = 10?
In 5 years, there will be 10 open stores.
independent variable
Input factor; Has a direct effect on the other variable
Norman is responsible for converting incoming foreign profits to American currency. On a given day, Norm knows that one U.S. dollar is equivalent to 0.8 euro. Of the two possible functions in this scenario, identify the input and output variables of the function that should most logically be labeled the original function, and write the corresponding equation.
Input: Profit in euros Output: Profit in U.S. dollars The function is D(e) = 1.25e.
Which pair represents the inputs and outputs of a function?
Input: Social security number; Output: A person's name
Which represents the inputs and outputs of a function?
Input: Speed of car; Output: Distance traveled in one hour
A high-interest loan company uses the function L(d) = 100e0.02d to calculate late fees, L, for payments overdue by d days. What does L(67) represent?
L(67) is the amount due in late fees for a payment that is 67 days overdue, which would be $381.90.
1st Degree Polynomial:
Linear function, constant rate of change
Cooper is a counselor of summer camp activities. Cooper asked the children to group themselves by their favorite colors. The results are shown in the following table. Which input-output pair corresponds to the third row of the table? Assume that C predicts N.
N(Green)=3
A regression model has an 𝑟2- value of 0.955. Does this alone mean it is a strong model?
No. If there were only a few data points (say, three), then a high coefficient of determination is not meaningful and does not indicate a strong model.
How would outliers affect a regression's coefficient of determination?
Outliers would cause the coefficient of determination to be smaller (or further away from 1).
The population of the United States in millions of people can be estimated with the function P(t) = 282(1.01)t, where t = 0 represents the population, P, in the year 2000, and t is measured in years. What does P(-2) represent?
P(-2) represents the population in 1998, which was approximately 276 million people.
The profit for a manufacturer of a certain product is given by the function P(x) = -4x3 + 12x2 + 16x, where P is the total profit (in thousands of dollars) for x items produced (in thousands). What is the average rate of change from x = 4 to x = 8?
Profit decreases by $288,000 per 1,000 items produced.
in Lily's small firm, every 5 additional employees add $10,000 to monthly revenue, which is presently estimated at $25,000. Assuming the effect of adding employees is stable, write a function, 𝑅(𝑒), to model the company's monthly revenue in dollars, where e is the number of new employees to be hired.
R(e)=2,000e+25,000
In average houses, there are 4 rooms and an additional room per household members. In larger houses, there are 7 rooms and an additional 2 rooms per one household member. Which function, R(p)𝑅(𝑝), describes the number of rooms in a larger house, where p represents the number of people in the household?
R(p) = 2p + 7
Interval:
Range of numbers between its endpoints
14. A company has the following information on last quarter's sales (month 7 refers to July): Month Sales (in thousands of dollars) 7 303 8 328 9 333 What was the average rate of change over the entire quarter (from July to September)?
Sales increased an average of $15,000 per month.
A cable company, a satellite dish company, and a streaming service all offer TV viewing plans based on the number of channels, c, on the plan. The price of each plan is given below: Cable: P(c) = 3c + 60 Satellite dish: P(c) = 4c + 35 Streaming service: P(c) = 5c + 50 Which company's cost per additional channel is most expensive?
Streaming service
Given the inverse function T(c)=−2c+50𝑇(𝑐)=−2𝑐+50, what is 𝑇(15), and what does it mean?
T(15) = 20; it implies that if the company keeps 15 computers, Carlo can purchase 20 new tablets.
The graph below shows the three-month average percentage year-on-year growth for real total pay for the years 2010-2016, along with the Office for Budget Responsibility's (OBR's) growth forecast for the years 2016-2021. What can be concluded from this graph?
The OBR forecasts the growth will remain above 1% between 2016 and 2021.
A bank uses the function A(t) = 1000(1.023)t to calculate an account balance, A, after t years. What is the average rate of change of the account balance from t = 2 to t = 12?
The account will increase by $26.72, on average, each year.
An airplane was cruising at an altitude of 39,000 feet and descended as shown in the table below. Descent of Airplane Altitude (ft.). Time (min.) 39,000 0 33,000 10 27,000 20 21,000 30 15,000 40 9,000 50 3,000 60 What can be concluded from this table?
The airplane had a descent rate of 600 ft./min.
In addition to Celsius and Fahrenheit, there is one additional unit of measure used in temperature called Kelvin. Temperature in Kelvin can be calculated by adding 273.15 to the temperature in Celsius. What is a correct statement about this relationship?
The function to convert Kelvin to Celsius is 𝐶(𝐾)=𝐾−273.15
The price of gasoline is $2.50 per gallon. The cost, C, can be represented by the function 𝐶(𝑛) = 2.5𝑛, where n is the number of gallons of gasoline. What is the inverse function?
The inverse function gives the number of gallons of gasoline as a function of the cost when the price of gasoline is $2.50 per gallon.
Rona, a real estate agent, is looking to sell a home. She has three potential buyers, and she knows how many people are in each family. All three families are eager to see the house, and Rona knows that the first to see it will probably make an offer. Which function—the original or its inverse—would be more useful to Rona in determining which family should get the first viewing?
The inverse function would be more useful here. Rona knows the number of rooms in the house, so she can calculate the number of people it will fit, then match it to the number of people in each family to decide which family would best "fit" the house and give that family the first viewing.
Crime starts increasing in a neighborhood. In response, the local police station increases the number of patrols they do in the neighborhood. Which variable is the independent variable in this context?
The number of crimes
The number of law enforcement officers has increased in a neighborhood. People later notice that the number of crimes has decreased. Which variable is the independent variable in this context?
The number of law enforcement officers
You are an IT manager with six people on your team. Each team member works 40 guaranteed hours a week. If a new software tool or platform is rolled out, more help is needed for customer support, and your team members may work a few hours of overtime. Which of the following choices listed is a variable?
The number of overtime hours
The output of toy cars in a toy factory is affected by the number of employees working per shift. Which variable is the dependent variable?
The number of toy cars produced
Dependent Variable
The outcome factor; the variable that may change in response to manipulations of the independent variable.
You work in IT on many different platforms (PC, Mac, Linux, etc.) with different types of software and tools. One of your friends also works in IT. She is the administrator of a college learning management system. Which choice is a qualitative variable?
The platform you work on for any given project
A rabbit population can be estimated with the function P(t)= 117/1+5e^-0.2t, where t = 0 represents the rabbit population, P, in the year 2010, and t is measured in years. What is the average rate of change of the rabbit population from 2020 to 2040?
The rabbit population is increasing by about 2.29 rabbits per year during this time.
Tyrelle runs an IT department of a small company and currently has 30 computers. Each member on her team needs a tablet and a computer. So, she can purchase just tablets until she needs more than 30. Then she must purchase a tablet and a computer for her employees. Tyrelle installed a remote operating software on both devices. The total number of computers is represented by C, and T represents the number of tablets purchased. Which of the following correctly represents a relationship between the number of computers and number of tablets with a valid, real-world application for that function?
The relationship could be represented by 𝐶(𝑇)=𝑇−30. You could use this function to determine the number of computers Tyrelle would need if she were to purchase 50 tablets.
An IT firm generates the following numbers to estimate its revenue in thousands of dollars based on the number of employees working on a project. EmployeesRevenue (in thousands) 10 $12.40 20 $21.99 30 $30.74 40 $36.01 50 $38.43 Which conclusion can be made by examining the average rate of change between consecutive rows?
The revenue is increasing at slower and slower rates in relation to the number of employees.
Which of the following is a predicted output based on an input?
The temperature of the refrigerator based on the selected setting.
The heat on a stove brings a pot of water to a boil. Which variable is the independent variable?
The temperature of the stove
The salvage or depreciated value, V, of a new truck purchased for $58,000 is given by the linear function V(t) = 58,000 - 4,000t, where t is the time in years between 0 and 10. What is the correct interpretation of V(3)?
The truck's depreciated value after 3 years is $46,000.
A big box store found its number of customers, C, in thousands on Saturday is predicted by the time of day in hours, t. In the graph, t = 0 corresponds to midnight of the current day and t = 24 corresponds to midnight of the next day. How should the average rate of change from t = 8 to t = 10 be interpreted?
There are, on average, 1,000 more customers per hour.
Which variable is a qualitative variable?
Your shirt color
Which choice is a variable?
Your weight
Trends:
a general direction in which something is developing or changing
2nd Degree Polynomial:
a quadratic polynomial in the shape of a parabola (upward or downward) and has a variable rate of change.
For every person on your team, t, you expect to get 4 minor projects a month, p. Which function input-output notation represents the situation where you have 3 people on your team?
p(3) = 12
Jenice is hiring a babysitter for the evening. She pays $5 to cover the sitter's transportation costs, and $10 per hour of babysitting. Jenice needs her babysitter for four hours this Saturday night. Which of the following is the function that represents Jenice's babysitting costs, where x is the number of hours the babysitter works?
𝐶(𝑥) = 10𝑥 + 5
What is the inverse of f(x)?
𝑓^−1 (𝑥)
