COLLEGE ALGEBRA. Linear Functions & Systems of Linear Equations

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1 COLLEGE ALGEBRA By: Sister Mary Rebekah Cornell-Style Fill in the Blank Notes and Teacher s Key Linear Functions & Systems of Linear Equations 1

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3 Slope & the Slope Formula main ideas & questions Slope notes & examples Slope is the between the points. Think rise. Slope is denoted by the letter m. It is a run constant, which should always be written as a coefficient to the independent variable (usually x or t), when written in slope-intercept form. What is the slope of the following linear functions? 1. y = 1 2 x h = 1.06t a n = 8 n x 1 3 y = 1 6. y = x = 16 Slope Formula m = Directions: Find the slope for the following linear situations. 8. A line that passes through the points (113, 735) and (163, -515). (y 2 -y 1 ) (x 2 -x 1 ) 9. A line that passes through (2, 73,950) and the y-axis at 76, A line the passes through the y-axis at 17 and the x-axis at Every 5 minutes, Marci drive 7.5 miles. Sister Mary Rebekah 3

4 Slope as Rate of Change main ideas & questions Constant Rate of Change notes & examples Slope can also be thought of as. What does that mean? Interpreting Slope in Real World Situations Directions: Interpret the following linear functions. College Enrollment. The total fall enrollment in 4-year state institutions is given by y = 0.014x , where x is the number of years after 1990 and y is the millions of students. a) Is this a linear function? b) Why or why not? c) What is the independent variable? d) What is the rate of change? e) What is the year at t = 0? f) What was the total fall enrollment in 2000, according to this model? Chemistry. The table shows the temperature of a solution after it has been removed from a heat source. Time (min) a) Is this a linear function? Temperature ( ) b) Find the rate of change in the temperature for the solution. c) Write a linear equation, which would be an accurate model for this data. d) According to the model, what would be the temperature of the water at 14 minutes? e) According to the model, when would the temperature of the water be room temperature? Sister Mary Rebekah 4

5 Heating Water/Rate of Change Lab DIRECTIONS 1. Turn on the hot plate to the dial straight up; turn on the stirring rod to level 1. Record the temperature of the water every thirty seconds for 5 minutes. t sec Temp C 0:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 4:30 5:00 2. Now place the beaker in an ice bath. Wait 1 minute. Then record the temperature of the water every thirty seconds for 5 minutes. t sec Temp C 0:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 4:30 5:00 3. Create a scatter plot and linear regression for the first experiment. Sketch the points and a trend line below. What is the possible equation for this line? Equation: Sister Mary Rebekah 5

6 4. Create a scatter plot and linear regression for the second experiment. Sketch the points and trend line below. What is the possible equation for this line? Equation: 5. Please discuss the rate of change for each of the two experiments. Write an intelligent response of how this experiment demonstrates the validity of the formula for slope/rate of change. Sister Mary Rebekah 6

7 Graphing Linear Equations main ideas & questions Point-Slope Form notes & examples Directions: Write a linear equation given two points. 1. (-6, 1) and (-3, 7) 2. (-2, -2) and (-6, 8) Slope- Intercept Form Directions: Write a linear equation in slope-intercept form given a point and its slope. 3. (-1, 7); m= (3, -1) Standard Form Directions: Write a linear equation in standard form given the following lines. 5. A line that passes through the y-axis at 4 and has a slope of -1/2. 6. (0,6) (4,0) Sister Mary Rebekah 7

8 Graphing Linear Equations 7. Graph y = 2x Graph y = 3x Graph 4x y = 8 x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: 10. Graph 5x 4y = Graph 3 4 x 1 3 y = Graph 5x = 2y x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: 13. Graph x 2y 2 = Graph f x = 5 6 x Graph 4x 8 = 9y 5 x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: x-intercept: y-intercept: slope: Sister Mary Rebekah 8

9 Special Linear Functions main ideas & questions Constant Functions Identity Function Direct Variation Functions notes & examples 1. Graph y=5 2. Graph y=5 3. Graph y=x 4. Graph x=5 5. Graph y=3x 7. Presume that x and y vary directly. If y= -7, when x= -28, find y when x = 20. H O Y V U X 6. Graph y=-1/2x Inverse Variation Sister Mary Rebekah 9

10 Direct Variation Functions F inding the constant (k) Identify the constant of the ordered pairs below. Then, write the equation to represent the relationship. 8) 1,4, 2,8, 3,12, (4,16) 9) ( 6,3), 4,2, 0,0, (2, 1) 10) 11) x y x y Identifying the equations Identify the equations below that represent a direct variation. If yes, identify the constant of variation. 12) y = 3x 13) y = 4 5 x 14) y = 2 15) 2y = 5x 16) y = x 4 17) 5x + y = 0 F inding missing values If the following ordered pairs represent a direct variation, find the missing value. 18) (2, 4) and ( 6, y) 19) (4,16) and (x, 24) 20) (x, 16) and (6,24) 21) If y = 10 when x = 4, find y when x = ) If y = 80, when x = 32, find x when y = 100. Sister Mary Rebekah 10

11 Parallel and Perpendicular Lines main ideas & questions Parallel Lines notes & examples Parallel lines have the. 1) Write the equation of a line through (4,5) and parallel to the line with the equation 3x + 2y = 1. Graph both equations. Perpendicular Lines Perpendicular lines have the. 2) Write the equation of a line through (4,5) and perpendicular to the line with the equation 3x + 2y = 1. Graph both equations. Examples Write the equation of a line in slope-intercept form, which fits the following conditions. 3) Parallel to y = 5x + 5, passing through (6,-2). 4) Parallel to the line y = 3 x 2, 5 passing through (0,2) 5) Perpendicular to y = 5x + 5, passing through (6,-2). 6) Perpendicular to the line y = 3 x 2, passing through (0,2) 5 7) Write the equations of two lines, which are parallel to each other and both of which pass through the origin. Sister Mary Rebekah 11

12 Average Rate of Change main ideas & questions Average Rate of Change notes & examples Up to this point we have been using rates of change which are mostly constant. Many real-world situations, however, involve rates of the change that are not constant. These situations are often described using and average rate of change over a specified interval. Average Rate of Change Formula 8) Music. Refer to the graph below. Find the average rate of change of the percent of total music sales for both CDs and downloads from 2001 to Compare the rates. a) Calculate the Average Rate of Change for CD sales: b) Calculate the Average Rate of Change for Download sales: Examples How is the average rate of change over some interval of points related to the slope of a line connecting the points? For any function, the average rate of change between two points on its graph is the slope of the line joining the two points. Such a line is called a secant line. 9) y = x 2 x + 1; [0,3] 10) y = 1 x 2 ; [ 3, 2] 11) Find the average rate of change of f(x) = x 2,over the interval [-8,-7] 12) Find the average rate of change of g(x) = 6x^2 + 7, over the interval [11,12]. Sister Mary Rebekah 12

13 Average Rate of Change 13) Tesla Automobile Sales. Sales and projected sales for the electric-powered Tesla automobile are shown in the figure below. The number of Tesla sales can be modeled by the function S t = 3.521t t where S(t)is on the thousands and t is the number of years after A graph is of S(t) is shown below , ,000 50, , Years after 2010 Sales a. Find the average rate of change of S(t) between (2, S(2)) and (6, S(6)). b. Interpret the average rate of change in terms of this situation. a. What is the relationship between the slope of the secant line joining the points (2, 1.896) and (6, 99.28)

14 Systems of Linear Equations main ideas & questions What is a System of Linear Equations? notes & examples A situation in which two or more linear equations are involved is called a system of linear equations when the equations have the same variables. A system of equations can have exactly o n e solution, n o solution, or i n f i n i t e l y m a n y solutions. A solution to a system of equations in two variables is an ordered pair that satisfies b o t h equations in the system. We will solve systems of linear equations in two variables by g r a p h i n g, by s u b s t i t u t i o n, and by e l i m i n a t i o n method. 1) The figure below shows that the size of China s manufacturing sector exceeded that of the United States in this decade. If we find the linear functions that model these graphs, with x representing the number of years past 2000 and y representing the sizes of the manufacturing sector in trillions of 2005 dollars, the point of intersection of the functions will represent the simultaneous solution of the two equations because both equations will be satisfied by the coordinates of the point United States China 0.5 Solve by graphing When solving a system of linear equations by graphing, graph each equation, then determine its intersection. y = 300x ) ቊ y = 5800x x 4y = 21 3) ቊ 2x + 5y = 9 Solve algebraically SOLVE BY SUBSTITUTION y = 10 4) ቊ 4x 2y = 6 SOLVE BY ELIMINATION 3x 4y = 10 5) ቊ 4x 2y = 6 6) Suppose the graphs by the previous example on the Chinese and American manufacturing sector can be modeled by the functions y = 0.158x y =.037x Sister Mary Rebekah 14

15 Systems of Linear Equations-Word Problems Directions: For each problem, a) define your variables, b) set up a systems of linear equations, c) and solve. 1. Landon babysits and works part time at the water park over the summer. One week, he babysat for 3 hours and worked at the water park for 10 hours and made $109. The next week he babysat for 8 hours and worked at the water park for 12 hours and made $177. How much does Landon make per hour at each job? 2. Kent has a collection of pennies and nickels with a value of $1.98. The number of pennies he has is five less than twice the number of nickels. How many of each coin does Kent have? 3. At the fast food restaurant, four cheeseburgers and five small fries have a total of 2,310 calories. Three cheeseburgers and two small fries have a total of 1,330 calories. How many calories does each item contain? 4. One month, a homeowner used 150 units of gas and 520 units of electric for a total cost of $ The next month, 210 units of gas and 405 units of electric were used for a total cost of $ Find the cost per unit of gas and electric. 5. Kym is renovating the first floor of her home. She bought 750 square feet of laminate flooring and 525 square feet of carpet and paid $ She went back to the store and bought an additional 100 square feet of laminate flooring and 75 square feet of carpet and paid $ Find the cost per square foot of each type of flooring. Sister Mary Rebekah 15

16 More Systems of Linear Equations Applications 6. Courtney has a total of 88 stamps worth $ Some are 25 stamps and some are 2 stamps. How many of each does she have? 7. Tom put 18 gallons of midgrade gas in his truck and filled up his empty five gallon gas tank with regular gas for his lawnmower at home. He spent $ The following week, he put 14 gallons of midgrade gas in his truck and topped off his five gallon tank with just one gallon of regular gas. If he paid $39.75 and the prices remained the same, find the price per gallon of midgrade and regular gas. 8. A total of $12,000 was invested in two types of bonds. One pays 8% simple interest while the other pays 10.5%. Last year, the annual interest earned on the two investments was $1,145. How much was invested at each rate? 9. A collection of dimes and quarters is worth $9.55. If the quarters were dimes and the dimes were quarters, the total value would be $7.60. Find the number of each coin. 10. A 2018 Honda accord costs $25,100 (msp), but the cost in gas per month is $430. A 2018 Honda accord hybrid costs $31,580 (msp), but the monthly gas bill is $250. At what point does the hybrid pay for itself? Sister Mary Rebekah 16

17 COST main ideas & questions Cost Fixed Cost notes & examples Variable Cost Marginal Cost Examples 1) For total cost, total revenue, and profit functions that are linear, the rates of change are called marginal cost, marginal revenue, and marginal profit respectively. Suppose that the cost to produce and sell a product is C x x = 6790, where x is the number of units produced and sold. This is a linear function, and its graph is a line with slope Thus, the rate of change of this cost function is called the ; it s $54.36 per unit produced and sold. This means that the production and sale of each additional unit will cost an additional $ ) Suppose the monthly cost for the manufacture of golf balls is C(x) = x, where x is the number of balls produced each month. a) What is the slope of the graph of the total cost function? b) What is the marginal cost (rate of change of the cost function) for the product? c) What is the cost of each individual ball that is produced in a month? Sister Mary Rebekah 17

18 Revenue & Profit main ideas & questions Revenue Marginal Revenue notes & examples Profit Example Marginal Revenue and Marginal Profit. A company produced and sells a smart phone with revenue given by R x = 89.50x, dollars and cost given by C s = 54.36x dollars, where x is the number of smartphones produced and sold. a) What is the marginal revenue for this smartphone, and what does it mean? a) Find the profit function. a) What is the marginal profit for this smartphone, and what does it mean? Sister Mary Rebekah 18

19 Revenue & Profit main ideas & questions Breaking Even notes & examples Break-Even. A company is said to break even from the production and sale of a product if the total revenue equals the total cost that is if R(x) = C(x). Because profit, we can also say that the company breaks even if the profit for the product is zero. Examples 1) Suppose a company has its total revenue for a product given by R = 5585x and its total cost is given by C = 61, x, where x is the number of thousands of tons of the product that are produced and sold per year. The company is said to break even when the total revenue equals the total cost that is, when R=C. Find the number of thousands of tons of the product that gives the break even and how much revenue and cost are at that level of production. 2) A manufacturer of reading lamps has a total revenue given by R=15.80x and a total cost given by C= x, where x is the number of units produced and sold. Find the number of units that gives break-even for this product. 3) A manufacturer of automobile air conditioners has a total revenue given by R=136.50x and a total cost given by C= , where x is the number of units produced and sold. Use a nongraphical method to find the number of units that gives breakeven for this product. Sister Mary Rebekah 19

20 Toyota Camry & Hybrid Project Due Project The Situation Your family wants to buy a new Toyota Camry and is considering buying the hybrid version if the money saved on gas will be enough to pay the extra cost for the hybrid. The Challenge(s) How many years will it take from saved gas money to pay for the extra cost for a hybrid? Could you ever save enough money on gas (compared to your current car) to pay the entire car payment? Question(s) To Ask Defining assumptions: How many miles will the car be driven each year? What is the car s miles per gallon (mpg)? What does a gallon of gas cost? Exploring the answer: What is being measured on the axis? What do the y-intercepts mean? What is the significance of where the two lines intersect? What does it mean when one line has higher y-values than the other line? Consider This What information will you need to create a system of equations representing the cost of each vehicle? The list of information they will need to know includes: The price of the cars The distance (in miles) the cars are driven The miles per gallon (mpg) each car gets The cost of gas For the price of the cars, I have provided two options. Both options ignore extra fees such as registration and tax. We want to keep it simple, so the image Toyota Camry LE vs. Toyota Camry Hybrid LE MPG (from the Toyota website) has the MSRP (Manufacturer s Suggested Retail Price) listed and you can use that. Toyota Camry vs. Toyota Camry Hybrid MPG Sister Mary Rebekah 20

21 Toyota Camry Sales Data For systems of equations, the cars prices will be the y-intercepts. What is the y-intercept for the Camry? What is the y-intercept for the Hybrid? Gas Cost Next you will need to find out each car s yearly gas costs. Many assumptions will need to be made here and should be agreed upon as partners: How many miles will the car be driven each year? Use the chart below (from the US Department of Transportation Federal Highway Administration) to agree upon a common value. Consider your age and your gender and decide as partners what value you will use for your equation. Then please circle it! What is the car s miles per gallon (mpg)? Miles per gallon with separate values for city and highway driving is above in the image Toyota Camry vs. Toyota Camry Hybrid MPG. You should agree upon a common value by first determining what portion of the time the car will be driven in the city or highway. For example, if it is 50% city and 50% highway, then 29/41 will give an assumed value of 35 mpg If it is 75% city and 25% highway it would be 32 mpg Please test TWO scenarios. You may use these two if you wish or determine other scenarios that might fit better with your driving habits. 21

22 What does a gallon of gas cost? Does anyone have a receipt from recently purchasing gas? I went Kanku s two weeks ago and filled up my Toyota Camry at $2.299 a gallon. You can use this value, or you may use your recent receipt. For your information: PLEASE ASSUME: Both cars use regular unleaded gas. Both cars have equal maintenance costs (which may not be true) THE END IN MIND In the end you are creating two systems linear equations, four equations total: Driving Senario 1: y camry = (cost per year) x + msp camry y hybrid = (cost per year) x + msp hybrid Driving Senario 2: y camry = (cost per year) x + msp camry y hybrid = (cost per year) x + msp hybrid Then determine the solution to each system and interpret those results in your write-up. 22

23 Toyota Camry Project Scratch Work Driving Scenario #1 (description): Dimensional Analysis for Cost of gas per YEAR Regular CAMRY HYBRID CAMRY Driving Scenario #2 (description): Dimensional Analysis for Cost of gas per YEAR Regular CAMRY HYBRID CAMRY