Notes and Practice Packet - Loudoun County Public Schools · 1. The number s of tablespoons of sea salt needed in a saltwater fish tank varies directly with the number w of gallons

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Chapter 5 – Notes and Practice Packet

Point-Slope Form

You need two things: Write the equation of a line in point-slope form

1. m = -3, (3, 2) 2. m = 2, (-5, 1) 3. (-2, 1), (-3, 4) 4. (-4, -4), (1, -1) 5. (-3, -4), (0, -3) 6. (1, 1) (0, 2) 7. 8. 9. 10. Take all the point-slope forms and turn them into slope-intercept.

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Notes: Writing Equations in Slope-Intercept Form

Slope-Intercept Form: m - b – Write the equation if a line in slope-intercept form given the following:

I. Given the slope and the y-intercept: m = -3, b = 2

1. b = -1, m = -2 2. m = ½ , (0, 1) II. Given a graph: 3. 4. 5. III. Given a different equation: get y by itself 6. 3x – 7y = 14 7. 3x + 2y = 6 8. -3 = x – ½ y 9. y – 2 = 3(x – 1) 10. y + 3 = -2(x + 1) 12. y – 1 = ½ (x + 4)

IV. Given two points

13. (-1, 4) and (2, -5) 14. (2, -2) (4, -3)

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Notes: Standard Form The third form of writing equations is:

Standard Form

Rule 1: No fractions (multiply by the LCM of the denominator)

13

2y x

13 4

3x y

1 14

2 3x y

1 1 1

4 2 3x y

Rule 2: x and y have to be on the same side

Rules 3: A must be positive

How do you write standard form with just two points? (3, -3) (6, -2)

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Notes: Parallel and Perpendicular Lines Graph the following lines: y = ½ x – 4 m = x – 2y = -4 m =

y = 3x + 1 m = y – 4 = 3(x + 2) m =

Parallel Lines ________________ _____________________

Parallel Lines have the ____________ _____________ y = -2x – 4 m = x – 2y = -4 m =

y = 3x + 1 m =

y – 4 = 1

3 (x + 2) m =

Perpendicular Lines create a ____________ ____________ _________________

Perpendicular Lines have _____________ _______________ ________________ Given the following slopes, find the slope of a line that is PERPENDICULAR

4 -2 ½ 4

3 5

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Notes: Line Best Fit and Linear Regression Graph the following points

The Line Best Fit is a line that will help us predict additional values. Find two points where the line between them appears to follow the same correlation as the points. Write the equation in slope-intercept form between those two points. Predict the value of y when x = 10

Linear Regression Linear Regression is a process for your calculator to find the exact line best fit One the Calculator: Stat Edit L1… Enter all your x values in L1 and all your corresponding y values in L2. Make sure all other lists are empty. 2nd Quit Stat Calc 4 Enter Y = ax + b just like y = mx + b Plug in values where they go.

a) Make a scatter plot of the data. b) Draw a line of fit. c) Write an equation for the line. d) Estimate x = 10

X 0 1 1 2 4 5 6

Y 2 3.5 2.5 3.5 4 5 5

X -3 -3 -2 0 0 1 2 1 4 y -2 -1 0 -1 0 2 2 3 4

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Homework #5: Linear Regression Draw a line of fit for the scatter plot. Write an equation for the line. Use the Calculator to find the Line of Regression Write an equation for the line. Estimate x = 10

8.

X –2 –1 2 2 3 4

Y –0.5 –1 –1.2 –1.5 –1.5 –1.8

9. x –2 –1 0 1 2 3

y 4 2 1 –2 –1 –2

10. x 0 0 0.5 1.5 2 2.5 y –4 –3 –1.5 1 3 4

11. x –3 –2 –1 0 1 2

y 1 –1 0 –2 –4 –5

12. x 0 4 3 2 1 0 y –3 –2 0 –1 1 1

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Notes: Direct and Inverse Variation Direct Variation Your uncle tells you that he will pay you $10 per hour for yard work on the weekend. How much money will you have earned after 3 hours? after 5 hours? after 9 hours?

Direct Variation Equation Remember Two Things:

y kx

1. Tell if the equation represents a direct variation. If it is a direct variation, tell what the constant of variation is.

a. 3 b. 2 0 c. 8 2 0

2. Graph the direct variation

3. The graph of a direct variation is shown. a. Write the direct variation equation. b. Find the value of y when x = 6

4. The number (n) of egg cartons needed at a chicken farm varies directly with the number of eggs (e) laid each day. One egg carton can hold 12 eggs.

A. Write a direct variation equation that relates n and e. B. How many cartons are needed to hold 2880 eggs? C. How many eggs do 32 cartons hold?

Inverse Variation: The phrase “y varies inversely as x” uses the equation

ky

x

where k is the constant of variation.

Follow the same steps as direct variation:

Plug in what you know to find k to write the equation

Solve for whatever other value you are looking for

1. If y and x vary inversely, If y is 6 when x is 3

a. Write the inverse equation. b. What is y when x is 9? c. What is x when y is 2?

2. IF y and x vary inversely, if y is 4 and x is 2,

a. Write the inverse equation b. What is y when x is 1? c. What is x when y is 4?

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3. The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t. If it takes 2 hours for a square inch of ice to melt at 65º, find the constant of proportionality. How long will it take to melt if the temperature is 40?

Practice: Direct and Inverse Variation

1. The number s of tablespoons of sea salt needed in a saltwater fish tank varies directly with the number w of gallons of water in the tank. A pet shop owner recommends adding 100 tablespoons of sea salt to a 20 gallon tank.

a. Write a direct variation equation that relates w and s. (Find the constant of variation!) b. How many tablespoons of salt should be added to a 30 gallon saltwater fish tank?

2. The amount of vacation v (in hours) that an employee earns at Toys R Us varies directly with the amount of time t (in weeks) he or she works. Kaitlin worked 2 weeks and earned 3 hours of vacation.

Write a direct variation equation that relates v and t.

How many hours of vacation can Riza expect to earn if he works for 8 weeks?

3. Brenden is landscaping and plans to spread a layer of stone on a path. The number s of bags of stone needed vary directly with the depth d (in inches) of the layer. Jonathan used 10 bags to spread a layer of stone that is 2 inches deep.

Write a direct variation equation that relates s and d.

How many bags will Dillon need to spread a layer that is 3 inches deep?

4. The distance d (in meters) you travel on a bicycle varies directly with the number r of revolutions that the rear tire completes. Hannah travels about 2 meters on a mountain bike for every revolution of the tire.

Write a direct variation equation that relates d and r.

How many meters would Hannah travel on her bike after 1500 tire revolutions?

5. At a recycling center, computers and accessories can be recycled for a fee f based on direct variation with weight w. Holly has to pay $7.50 to recycle 30 pounds of accessories.

Write a direct variation equation that relates f and w.

How much would Leyla have to pay to recycle an 18 pound computer?

How much would Samantha have to pay to recycle a 10 pound printer?

6. A local newspaper charges by the word for printing classified ads (the cost directly varies with how many words are printed). A 14 word ad costs $5.88. How much would a 21 word ad cost?

7. In kick boxing, it is found that the force, f, needed to break a board, varies inversely with the length, l, of the board. If it takes 5 lbs of pressure to break a board 2 feet long, how many pounds of pressure will it take to break a board that is 6 feet long?

8. A theater company plans to hire people to build a stage set. The work time t (in hours per person) varies inversely with the number p of people hired. The company estimates that 25 people working for 300 hours each can complete the job. Find the work time per person if the company hires 30 people.

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Writing equations applications 1. Landscape Supply A landscape supply business charges $30 to deliver mulch. The mulch costs $23 per cubic yard.

a. Write an equation that gives the total cost (in dollars) of having mulch delivered to a site as a function of the number of cubic yards ordered.

b. Find the cost of having 8 cubic yards of mulch delivered to a site.

2. Cable Television A cable company charges $44 per month for basic service. Each premium channel costs an additional $16 per month.

a. Write an equation that gives the total cost (in dollars) of cable each month as a function of the number of premium channels.

b. Explain how you can use the equation from part (a) to approximate how many premium channels you can have for $80 a month.

3. Laser Printer A laser printer has a “sleep” mode that is an energy-saving feature. When a job is sent to the printer, it takes 45 seconds for the printer to warm up and then the printer prints pages at a rate of 6 pages per minute.

a. Write the time it takes the printer to warm up in minutes.

b. Write an equation that gives the total amount of time (in minutes) it takes the printer to warm up and print a job as a function of the number of pages in the job.

c. Find out how long it takes the printer to print a 50-page job if it must first warm up.

d. 4. Lunch Duty Everyone at camp takes turns being on lunch duty. You and your friend are in charge of making

sandwiches. You both can make 1 sandwich in 2 minutes. Your friend arrives 10 minutes earlier than you and starts making sandwiches.

a. Write equations that model the number of sandwiches made as a function of the number of minutes it takes you and your friend to each make sandwiches.

b. How many sandwiches will each of you make in 20 minutes?

c. How are the graphs of the equations from part (a) related? Justify your answer.