Name PD Linear Functions
Finding the Slope of a Line The steepness of the line is the ratio of rise to run, or vertical change to horizontal change, for this step. We call this ratio the slope of the line. Slope is also known as the rate of change. To determine the slope of a line, ou need to consider the direction, or sign, of the vertical and horizontal changes from one point to another. Example : Tr-It! a. b.
Speed (mi/h) Example : Tr-It! a. b. What is the rate of change of the car s speed from 0 to seconds? 60 0 0 0 0 0 0 0 0 x Time (s) Example : Special Slopes
Example : Positive Slope from a Table Tr-It! a. b. c. Example : Negative Slope from a Table Tr-It! a. b. Example 6: Find the slope of each line that passes through each pair of points The points (, ) and (-, 0) lie on a line. Tr-It! a. A (, -) and B (, ) b. C (-, ) and D (, )
c. J (-, 8), K (-, ) d. P (,-7), Q (-,-7) Slope describes the steepness of a line, also known as the rate of change. Slope of a line = change in - coordinates change in x - coordinates or rise run Possible graphs:. Positive Slope. Negative Slope. Zero Slope. Undefined Slope Parts of a Linear Equation slope Graph rise run Table change in change in x
. Graph the following situation: A swimmer climbs a ladder for two seconds to a waterslide that is five feet high. She sits for two seconds at the top of the slide, and then slides 8 feet down the slide into the water in two seconds. She staed stead at the same position underwater for two seconds before rising to the surface of the water in one second. She remained in the pool for one more second. Elev(ft) 6 6 7 8 9 0 Time (s) 6 a. What was the average rate of change for the swimmer going up the ladder? b. At what interval of time is the slope negative? c. Determine when the slope is zero.
. Determine the slope from the table. X 6 7 Y - 0 Then plot the points on the graph and find the slope. 8 7 6 8 7 6 6 7 8 x 6 7 8 Make a comparison between the slope from the table and the slope from the graph.. Use the tables to find the slopes: x x 0 0 6 8 0 6 8-6 6-6
State whether the slope is positive, negative, zero, or undefined. Calculate the slope to confirm our statement... 6. Find the slope of the line that passes through each pair of points. 7. J (-, 6), K (-, ) 8. P (,-7), Q (-,-7) 9. M (7, ), N (-, ) 0. A (0, 9), B (, ) 7
Y-Intercept Notes Finding the -intercept of a line -intercept is the coordinate of the point where a line crosses the -axis, it s also the initial value when x = 0. Example : Tr-It! a. b. Example : Determining the -intercept of a table a. b. c. Tr-It! a. b. 8
Example : Slope-Intercept Form of Linear Equations: = mx + b (m stands for slope and b stands for -intercept). = x + 7 slope -intercept. = x 8 slope -intercept. = 6 7x slope -intercept. = 0x slope -intercept. = slope -intercept 6. Given slope of 9 and the -intercept is. 7. Given slope of 0 and the -intercept is. 8. Given m and b 6. 9. Given m and b. 0. Given m and the -intercept is (0, ).. Given slope of 0. and b 0. 6 9
Practice. Use the tables to find the -intercept: x x 0 0 6 8 6 0 6 9-6 6 - Use the graphs to determine each -intercept.... 0
Find the slope and -intercept of the line represented b each equation:. = 6x + slope -intercept 6. = -x slope -intercept 7. = 7 9x slope -intercept 8. = -x slope -intercept 9. = 8 slope -intercept Use the given information to write a linear equation in slope-intercept form: 0. Slope is and the -intercept is -6.. Slope is 7 and the -intercept is -.. m 8 and b.. m 0 and b.. m and the -intercept is (0, ).. Slope is and b 0
Notes-Graphing a Line Warm-Up!. For a homework assignment, Sarah must draw a line passing through the points (-, -) and (, ). Graph Sarah s line on the grid below. x. Graph a line that goes through the following points:,,, x Graph a linear relationship based on information provided:. Given slope of and the -intercept is. x. Given slope of 0 and the -intercept is. x
. Given m and the -intercept is (0, ). x. Given m and the point (0, -). Graph the linear equations: (Hint: identif the slope and -intercept). Graph: x.. Graph: x. x x x
. Graph: x. x x x Graphing Equations b Making Tables:.
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Practice. Graph a line that goes through the following points: (0, 0) and (, ) x. Graph a line that goes through the following points:,,,0 x 6
Graph a linear relationship based on information provided:. Given m and the -intercept is (0, -).. Given slope of and b 0 x x. Using the equation: x, complete the table below and then graph the equation. x - 8 7-6 0 8 7 6 6 7 8 x 6 7 8 7
Graph the linear equations: (Hint: identif the slope and -intercept) 6. Graph: x. 7. Graph: x x x 8. Graph: x 9. Graph: x x x 8
Notes-Writing Linear Equations from a Graph Use the graph to determine the linear equation..... 9
Use the table to determine the linear equation in slope-intercept form. Plot the points and use the graph to determine the linear equation in slope-intercept form. Make a comparison between the equation from the table and the equation from the graph.. x - - 7 - - 8 7 6 0 8 7 6 6 7 8 x 6 7 8. x - 6-8 - 0 8 8 7 6 8 7 6 6 7 8 x 6 7 8 0
Writing Equations based on a point and slope: Use the information provided to determine the linear equation in slope-intercept form.. Given m and the point (-, ). x. Given m and the point (-, 0). x. Given m and the point (-, -). x
Writing Equations based on points:. Graph a line that goes through the following points,,,, 8 7 6 and write the equation. 8 7 6 6 7 8 x 6 7 8. Graph a line that goes through the following points,,,, and write the equation. 8 7 6 8 7 6 6 7 8 x 6 7 8
Practice Use the graph to determine the linear equation.... Writing an Equation From a Table: Use the table to determine the linear equation in slope-intercept form. Plot the points and use the graph to determine the linear equation in slope-intercept form. Make a comparison between the equation from the table and the equation from the graph.. 8 7 6 Use the table to determine the linear equation in slope-intercept form. x. 6. x 8 7 6 6 7 8 x 6 7 8 0-0 6 8
Writing Equations based on a point and slope: Use the information provided to determine the linear equation in slope-intercept form. 7. Given m and the point (-, -). x 8. Given m and the point (0, ). x 9. Given m and the point (-, ). x
Writing Equations based on points: 0. Graph a line that goes through the following points,,,,8 8 7 6 and write the equation. 8 7 6 6 7 8 x 6 7 8. Graph a line that goes through the following points, 0,,, and write the equation. 8 7 6 8 7 6 6 7 8 x 6 7 8
Spiral:.. Find the slope of the line. Describe how one variable changes in relation to the other. a) ; the amount of water decreases b gallons ever minutes. b) ; the amount of water decreases b gallons ever minutes. c) ; the amount of water decreases b gallons ever minutes. d) ; the amount of water decreases b gallon per minute. 6
Notes-Writing Equations Finding Equations from a Given Slope and a Given Point Use the information below to write a linear equation. A. slope is and the line passes through the point (-, 6) B. slope is - and the line passes through the point (, 7) Tr It! Use the information below to write a linear equation. A. slope is and the line passes through the point (, 6) B. slope is - and the line passes through the point (, ) 7
Finding Equations Given Points: Use the information below to write a linear equation. A. (, ) & (, ) B. (6, -) & (-, ) Tr It! Use the information below to write a linear equation. A. (, 7) & (, -) B. (6, -) & (-, -) 8
Is the relationship shown b the data linear? If it is, model the data with an equation. x x x 6 - -7-0 - - - 7 0 8 9 Practice 9
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Notes-Word Problems Warm-up: Find the rate of change. Explain what the rate of change means for the situation. Equations for Linear Relationships Cars and trucks are an important part of American life and culture. There are nearl 00 million licensed drivers and 0 million registered passenger cars in the United States. To help people keep their cars clean, man cities have self-service car washes. At most self-service car washes, the charge for washing a car and the compan s profit depend on the time the customer spends using the car wash. To run such a business efficientl, it helps to have equations relating these ke variables. Getting Read Vocabular: Rate of change a comparison between two quantities that are changing; this is slope. Initial Value is like the -intercept, it s the value for doing nothing; ex. flat fee.
Tr It! Sudzo Wash and Wax charges customers $0.7 per minute to wash a car. Write an equation that relates the total charge c to the amount of time t in minutes. Pat s Power Wash charges $.00 per car to cover the cost of cleaning supplies, plus $0.9 per minute for the use of water spraers and vacuums. Write an equation for the total charge c for an car-wash time t. Equations for Linear Relationships. The Squeak Clean Car Wash charges b the minute. This table shows the charges for several different times. a. Explain how ou know the relationship is linear. b. What are the slope and -intercept of the line that represents the data? c. Write an equation relating charge c to time t in minutes.
. Euclid s Car Wash displas its charges as a graph. Write an equation for the charge plan at Euclid s. Describe what the variables and numbers in our equation tell ou about the situation. Equation: Describe:. Use the function in the table at the right. a. Identif the dependent and independent variables. b. Write a rule to describe the function. c. How man gallons of water would ou use for 7 loads of laundr? d. In one month, ou used gallons of water for laundr. How man loads did ou wash?
. A television production compan charges a basic fee of $000 and then $000 per hour when filming a commercial. a. Write an equation in slope-intercept form relating the basic fee and per-hour charge. b. Graph our equation. x c. Use our graph to find the production costs if hours of filming were needed.
Practice
7. Complete the table if necessar and then write a function rule for the table. 8. Find the rate of change. Explain what the rate of change means for the situation. 9. The function rule C 6 0n relates the number of people, n, who attend a small concert to the cost in dollars, C, of the concert. Evaluate the function to find the range for the domain values of 7, 9, and people attending. 6
Notes-Arithmetic Sequences Example Extending Number Patterns Use inductive reasoning to describe each pattern, and then find the next two numbers in each pattern. A),, 8,,, B),, 8, 6,, C),, 9, 6,, Tr-It! Use inductive reasoning to describe each pattern, and then find the next two numbers in each pattern. A), 9, 7, 8,, B) 9,,, 7,, C), -, 8, -6,, A number pattern is also called a sequence. Each number in the sequence is a term of the sequence. One kind of number sequence is an arithmetic sequence. You form arithmetic sequence b adding a fixed number to each previous term; this fixed number is called the common difference. Example Finding the Common Difference Find the common difference of each arithmetic sequence and then find the next two numbers in sequence. A) -7, -,,,, B) 7,, 9,,, 7
Tr-It! Find the common difference of each arithmetic sequence and then find the next two numbers in sequence. A),,, 7,, B) 8,, -, -7,, Example Writing a Function Rule for Arithmetic Sequence Consider the sequence 7,,, 9, think of each term as the output of a function. Think of the term number as the input. Term number: input Term: 7 9 output Let: n = the term number in the sequence A n value of the n th term of the sequence n A n 7 7 + 0 () 7 7 + 7 + () 7 + + 7 + () 9 n The general form of an arithmetic sequence A n a n d Tr-It! Write the function rule for the arithmetic sequence. A),,, 7 B) 8,, -, -7 8
Example Finding Terms of a Sequence Find the first, second, and twelfth terms of the sequence. Given: A n n Tr-It! Find the first, second, and twelfth terms of the sequence. A) A n n B) A n 6. n Practice Use inductive reasoning to describe each pattern, and then find the next two numbers in each pattern..,, -, -,, Find the common difference of the arithmetic sequence and then find the next two numbers in sequence.. -, -,,,, Write the function rule for the arithmetic sequence.. -9, -,, 6 Find the first, second, fifth, and ninth terms of the sequence.. A n 9 n 6 9
For & 6: Find the next two terms in the sequence, explain if it is an arithmetic sequence or not. If the sequence is arithmetic, write the function. If is not arithmetic, describe the pattern.., -,,,, 9 6. 7, 7, 7, 7,, For 7 & 8: Write the first five terms of the sequence, explain what the fifth term means in context to the situation. 7. A bab s birth weight is 7 lbs. oz., the bab gains oz. each week. 8. The balance of a car loan starts at $,00 and decreases $0 each month. 9. Find the second, fourth, and eighth terms of the sequence. A n n 0. The Fibonacci Sequence is,,,,, 8,, Find the next two terms in the sequence, explain if it is an arithmetic sequence or not. If the sequence is arithmetic, write the function. If is not arithmetic, describe the pattern. 0
Notes-Standard Form There are two common forms of a linear equation. When the values of one variable depend on those of another, it is most natural to express the relationship as = mx + b. Most of the linear equations ou have seen have been in this slopeintercept form. When it is more natural to combine the values of two variables, the relationship can be expressed as ax + b = c. This is the standard form of a linear equation. Wh do we need to think about this tpe of equation? --------------------------------------------------------------------------------------------------------------------- It is eas to graph a linear equation of the form = mx + b on a calculator. Can ou use a calculator to graph an equation of the form ax + b = c? Can ou change an equation from ax + b = c form to = mx + b form? ---------------------------------------------------------------------------------------------------------------------
Example : Connecting mx b and ax b c Four students want to write x + = 9 in equivalent = mx + b form. Here are their explanations: A. Did each student get an equation equivalent to the original? If so, explain the reasoning for each step. If not, tell what errors the student made. B. What are the steps to converting an equation into slope-intercept form? C. What does it mean for two equations to be equivalent?
Example : Graph the following equations. Converting standard form equations ( ax b c ) into slope-intercept form ( mx b ). a) x b) x 8 x x Tr It! a) x b) 600 x 0 c) x 0
Example : Converting slope-intercept form ( mx b ) into a standard form equation ( ax b c ). a) x b) x Tr It! a) x b) x c) x Practice Converting standard form equations ( ax b c ) into slope-intercept form ( mx b ).. x. 6x 9. x. x 8
. 7x 6 6. x Converting slope-intercept form ( mx b ) into a standard form equation ( ax b c ). 7. x 8. x 9. x 0. x 6. x 7. x
Notes-Standard Form Word Problems Equations With Two or More Variables You have done a lot of work with relationships involving two related variables. However, man real-world relationships involve three or more variables. For example, consider this situation: The eighth-graders are selling T-shirts and hats to raise mone for their end-of-ear part. The earn a profit of $ per shirt and $0 per hats. This situation involves three variables: the number of T-shirts sold, the number of hats sold, and the profit. The profit for the fundraiser depends on the number of hats and the number of T-shirts sold.. What equation shows how the profit, p, is related to the number of shirts sold, s, and the number of hats, h, sold?. Find the profit if the students sell: a) 0 shirts and 0 hats b) shirts and 0 hats. What ideas do ou have for finding solutions to the equation? 6
The equation relating p, s, and h represents ever possible combination of T-shirts, hats, and profit values for the fundraiser. Suppose the class sets a profit goal of P = $600. Finding combinations of T-shirt and hat sales that meet this goal requires solving an equation with onl two variables, s and h. Solving Equations With Two Variables a) Write an equation that relates the number of t-shirts and hats sold to a $600 profit. b) Find three pairs of solutions to our equation above; then graph the solutions below. c) Wh does the graph onl occur in the first quadrant? Explain. d) What do the coordinate pairs mean? Hint: in context to the word problem. Example : 7
Number of Hours Walking Students in Eric s gm class must cover a distance of,600 meters b running or walking. Most students run part of the wa and walk part of the wa. Eric can run at an average speed of 00 meters per minute and walk at an average speed of 80 meters per minute. a. If Eric runs for x minutes and walks for minutes, write an equation for the total distance that Eric will cover. b. Suppose Eric runs for minutes and walks for 8 minutes, will he reach the finish line? c. Find three pairs of solutions to our equation in part a. Practice Suppose that a participant in a 0-K run/walk event could run at a pace of about 0 kilometers per hour and walk at a pace of about 6 kilometers per hour. Write an equation that relates x and to the goal of covering 0 kilometers. b) Find three pairs of solutions to our equation above; then graph the solutions below. 0-K Run/Walk 0 9 8 7 6 6 7 8 9 0 Number of Hours Running c) Based on one of our ordered pairs, how does it relate to the situation. 8
. The Plano Texans are a outh marching band that competes with other groups all over the countr. The band director rents instruments to members. Each trumpet rents for $0 per month and each drum rents for $ per month. Write an equation that relates the trumpet and drum rentals if the band director charges $ in rental fees. Which of the situations below is accurate: The band director renting out 9 trumpets and 7 drums or 7 trumpets and 9 drums. Justif our answer. Notes-Intercepts If the -intercept is where the graph crosses the -axis, then the x-intercept is where the graph crosses the. Finding intercepts To find x-intercept:. Plug in 0 for.. Solve for x. To find the -intercept:. Plug in 0 for x.. Solve for. Tr this: Find the intercepts of x = 6 The Sandia Peak Tramwa in Albuquerque, New Mexico, travels a distance of about 00 meters to the top of Sandia Peak. Its speed is 00 meters per minute. The function f(x) = 00 00x give the tram s distance in meters from the top of the peak after x minutes. Find and interpret the intercepts for the situation. Find the intercepts of 8x + 7 = 8 A hot air balloon is 70 meters above the ground and begins to descend at a constant rate of meters per minute. The function f(x) = 70 x represents the height of the hot air balloon after x minutes. Find and interpret the intercepts for the situation. 9
We ve alread looked at slope-intercept form. Standard form looks like Ax + B = C where A, B, and C are numbers. Example: x + = A lake was stocked with 0 trout. Each ear the Graph the following line using intercepts: population decreases b. The population of trout 8 = x + 08 in the lake after x ears is represented b the function f(x) = 0 x. Find the intercepts and use them to graph the function. What does each intercept represent? The number of brake pads needed for a car is, and a manufacturing plant has 80 brake pads. The number of brake pads remaining after brake pads have been installed on x cars is f(x) = 80 x. Find the intercepts and use them to graph the function. What does each intercept represent? Graph the following line using intercepts: 7x = A tank is filling up with water at a rate of gallons per minute. The tank alread had gallons in it before it started being filled. Write an equation in standard form to model the linear situation. A pool that is being drained contained 8,000 gallons of water. After hours,,00 gallons of water remain. Write an equation in standard form to model the linear situation. 0
Your school sells adult and student tickets to a school pla. Adult tickets cost $ and student tickets cost $. The total value of all the tickets sold is $7000. Write an equation in standard form to model the linear situation. You are in charge of buing the hamburger and chicken for a part. You have $60 to spend. The hamburger costs $ per pound and chicken is $ per pound. Write an equation in standard form to model the linear situation. Some lines are vertical or horizontal. These have undefined and zero slopes but what do the equations of their lines look like??? And how do we graph them? Graph the following line: Graph the following line: x = = What is the equation of the following line? What is the equation of the following line? Practice Find the intercepts of each of the following situations:. 9x = 6 + 7. x + = 0 Find and interpret the intercepts.. The temperature in an experiment is increased at a constant rate over a period of time until the temperature reaches 0 C. The equation = x 70 gives the temperature in degrees Celsius x hours after the experiment begins.. At a fair, hamburgers sell for $.00 each and hot dogs sell for $.0 each. The equation x +. = 0 describes the number of hamburgers and hotdogs a famil can bu with $0.
Find the intercepts and use them to graph the function.. x 6 = 6. = x + Find the intercepts and interpret them for each situation. Use the intercepts to graph each situation. 7. The air temperature is -6 C at sunrise and rises C ever hour for several hours. The air temperature after x hours is represented b the function f(x) = x 6. 8. Connor is running a 0-kilometer cross countr race. He finishes kilometer ever minutes. Connor s distance from the finish line after x minutes is represented b the function f(x) = 0 x. Write an equation in standard form to model the linear situation. 9. A hot tub filled with 0 gallons of water is being drained. After. hours, the amount of water had decreased to 0 gallons. 0. A barrel of oil was filled at a constant rate of 7. gal/min. The barrel had 0 gallons before filling began.. A restaurant needs to plan seating for a part of 0 people. Large tables seat 0 people and small tables seat 6 people. (Let x represent the number of large tables and represent the number of small tables.) Graph the following lines.. x =. =
Write the equation of the following lines... 6. 7. Write a real-world problem that could be modeled b a linear function whose x-intercept is 6 and whose -intercept is 60. 8. Kim owes her friend $ and plans to pa $ per week. Write an equation of the function that shows the amount Kim owes after x weeks. Then find and interpret the intercepts of the function.
Notes-Comparing Functions Example : Three trains (A, B, and C) leave a train station at the same time. The graph shows the relationship between time and distance for Train A. Train B x. What is the slope of the graph?. What does this slope represent?. The relationship between time and distance for Train B is given b the equation above, where x represents hours and represents miles. Find the slope m.. Which train is moving faster, Train A or Train B? How do ou know?. The time-distance relationship for Train C is shown in the table above. What is the ratio of distance to time? 6. Compare the speed of Train C to the speeds of Train A and Train B. Example : The equation x represents the calories Jake burns when crosscountr skiing, where x is time in minutes and is the number of calories. The graph shows the calories he burns while plaing basketball. Which activit burns calories at a faster rate? Explain.
Example : A bowling alle offers different birthda part packages: Package A is represented b the function c 7 p where c is the total cost and p is the number of people. Package B is represented in the table. a) Compare the functions. Package B Number of People Total Cost $.0 9.00.0.00 b) If people attend the birthda part, which package will cost less and b how much? Example : Order the stocks from least to greatest rate of price increase. Alpha Inc. Week 0 Price $ 6.0 7.0 Beta Co. The starting price of $ decreases weekl b $.0. Delta Corp. 7w d (w is weeks, d is dollars)
Example : A. B. C. x x x - - - - 8 - -7-0 - - - - - - 0 0 0-0 - -6 0 9-8 -0 8 What are the patterns (compare/contrast)? Plot each of the previous tables on the graphs below, write equation for the linear functions: A. B. C. 6 6 6 6 6 x 6 6 6 x 6 6 6 x 6 Practice 6
. A scientist measured the growth rate of a bamboo plant at 6 inches in hours. She compared the growth rate of the bamboo plant to the growth rate of three other plants. Bull Kelp x 6, where x represents hours and represents inches. Did an of these plants have a faster growth rate than the bamboo plant? Explain our reasoning..the function m 0h, where m is the miles traveled in h, hours represents the speed of the first Japanese high speed train. The speed of a high speed train operating toda in China is shown in the table. a) Compare the functions. train? b) If ou ride each train for five hours, how far will ou travel on each Train Rate in China Hours Miles 7 6 7
. In the U.S. ou have man options for places to wash our car. Here are two different car wash locations: a) Compare the functions relating charge c to time t in minutes b) If ou wash our car for eight minutes, which car wash will cost less and b how much?. Which function has a greater rate of change? x 0 6-6 6 6 8 x. The functions below represent pricing plans for car rentals, where d is the number of das and c is the cost. Order the car rentals from least to greatest based on their rate of change. Avis d 7 c 00 0 0 60 Enterprise Total cost is $0 plus $ per da. Hertz 7d 0.c 0 8