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Chapter 7

Chapter 7. Linear Functions. EXPLORE! Relations & Functions. A RELATION expresses how objects in one group (inputs) are assigned or related to objects in another group (output).

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Chapter 7

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  1. Chapter 7 Linear Functions

  2. EXPLORE! Relations & Functions A RELATION expresses how objects in one group (inputs) are assigned or related to objects in another group (output). Look at the mapping diagram. It shows three possible relations. If each item, or input, has only one price, or output, then the relation is also a FUNCTION! The first two relations are functions because each item has only one price! The third relation is NOT a function because the binder has two prices. Copy the relation diagram. Draw lines from the input values to the output values so that the relation is a function. Do this again and draw lines so that the relation is NOT a function.

  3. Equations and Functions7-1-B Suppose you can buy magazines for $4 each. • Copy and complete the table to find the cost of 2, 3, and 4 magazines • Describe the pattern in the table between the cost and the number of magazines.

  4. Equations & Functions A RELATION is a set of ordered pairs. A FUNCTION is a relation in which each member of the input is paired with exactly one member of the output. A FUNCTION RULE is the operation (s) performed on the input value to get the output value. So, what were the ordered pairs in the previous table? Was this a function? What was the function rule?

  5. Equations & Functions The set of input values is called the DOMAIN. The set of output values is called the RANGE. A FUNCTION TABLE is where we are going to organize the input numbers, the output numbers, and the function rule. EXAMPLE!!! Jill saves $20 each month. Make a function table to show her savings after 1, 2, 3, and 4 months. Then identify the domain and range. Domain: {1, 2, 3, 4} Range: {20,40,60,80}

  6. Equations & Functions Try it Yourself! A student movie ticket costs $3. Make a function table that shows the total cost for 1, 2, 3, and 4 tickets. Then identify the domain and range. Functions are often written as equations with two variables to represent the input and output. Think back to Example 1 & look at this! Can you write an equation for the movie ticket problem?

  7. Equations and Functions An armadillo sleeps 19 hours each day. Write an equation using two variables to show the relationship between the number of hours h an armadillo sleeps in d days. h = 19d

  8. Examples A botanist discovers that a certain species of bamboo grows 4 inches each hour. Write an equation using two variables to show the relationship between the growth g in inches of this bamboo plan in h hours. Melanie read 14 pages of “Hound of the Baskervilles” each hour. Write an equation using two variables to show the relationship between the number of pages p she read in h hours. Anna earns $6.00 an hour working at a grocery story. Make a function table that shows Anna’s total earnings for working 1, 2, 3, and 4 hours. Then identify the domain and range.

  9. Function Tables Copy and complete each function table. Then, identify the domain and range. y = 0.5x y = 2/3x Self Assessment: Complete p. 380 # 1-4 all by yourself. Then check your work with a partner.

  10. More Relations and Functions7-1-B extra Independent Variables: Values (the Input) that are chosen and do NOT depend on the other variable. In an ordered pair, the x-value Dependent Variables: Values (the Output) that depend on the Input In an ordered pair, the y-value

  11. More Relations and Functions7-1-B extra Remember the last section? Determine whether the relation is a function. Explain. Another way to determine whether a relation is a function is to apply the VERTICAL LINE TEST. Use a pencil to represent a vertical line. Place the pencil at the left of the graph Move the pencil to the right. If for each value of x in the domain, the pencil passes through only one point on the graph, then the graph represents a function.

  12. More Relations and Functions7-1-B extra A function that is written as an equation can also be written in a form called FUNCTION NOTATION! y = 4xf(x) = 4x The function notation, f(x) is read “f of x” The variable y and f (x) both represent the dependent variable. Find f(3) if f(x) =5x Find f(4) if f(x)=2x Find f(-5) if f(x)= 20x Find f(-8) if f(x)= -5x + 10

  13. More Relations and Functions7-1-B extra Determine whether each relation is a function. Explain.

  14. More Relations and Functions7-1-B extra Determine a rule for each set of ordered pairs. Then copy and complete each function table. f(x) = x -3 f(x) = 4x -1 f(x) = 2x +6

  15. Functions and Graphs7-1-C The Westerville Marching Band is going on a year-end trip to an amusement park. Each band member must pay an admission price of $15. In the table, this is represented by 15m. Copy and complete the function table for the total cost of admission. Graph the ordered pairs (number of members, total cost) Describe how the points appear on the graph. The total cost is a FUNCTION of the number of band members. In general, the output y is a function of the input x. The graph of the function consists of the points in the coordinate plane that correspond to ALL the ordered pairs of the form (input, output) or (x,y)

  16. Functions and Graphs7-1-C The table shows temperatures in Celsius and the corresponding temperatures in Fahrenheit. Make a graph of the data to show the relationship between Celsius and Fahrenheit.

  17. Graph Solutions of Linear Equations Solution: consists of two numbers; one for each variable; make the equation true. Usually written as an ordered pair (x,y) Graph y = 2x +1 Create a table in which to organize your solutions Select any four values for the input (x) Fill in the table to find the output (y) and then the ordered pairs.

  18. Graph Solutions of Linear Equations Graph each equation. y = x – 3 y = -3x + 2 Create a table in which to organize your solutions Select any four values for the input (x) Fill in the table to find the output (y) and then the ordered pairs.

  19. Linear Function A function like y = 2x + 1 is called a LINEAR FUNCTION because its graph is a line. We already graphed this linear function earlier today. But if you draw a line through the points, you will have graphed ALL of the solutions.

  20. Linear Functions Michael Phelps swims the 400-meter individual medley at an average speed of 100 meters per minute. The equation d = 100t describes the distance d that Michael can swim in t minutes. Represent this function as a graph.

  21. Examples! Graph the function represented by this table which describes calories in fruit cups: Sandi makes $6 an hour babysitting. The equation m=6h describes how much money m she makes babysitting for h hours. Represent the function by a graph. Graph the equation: y= 3x - 1 Self Assessment: Complete p. 388 # 1-5 all by yourself. Then check your work with a partner.

  22. EXPLORE! Rate of Change Pampered Pets is a doggie daycare where people drop off their dogs while they work. Mrs. Bybee takes Layla to Pampered Pets several days a week. The table shows their prices. Use a graph to determine how the number of hours is related to the cost. Create a graph of this data! Is the graph linear? Explain What is the cost per hour, or unit rate, charged by Pampered Pets? Examine any two consecutive ordered pairs from the table. How do the values change. Is this relationship true for any two consecutive values in the table? Use the graph to examine any two consecutive points. By how much does y change? By how much does x change? How does this change relate to unit rate?

  23. Constant Rate of Change7-2-B The table shows Andi’s height at ages 9 and 12. What is the change in Andi’s height from ages 9-12? Over what number of years did it take place? Write a RATE that compares the change in Anei’s height to the change in age. Write this now as a unit rate. RATE OF CHANGE- a rate that describes how one quantity changes in relation to another. Usually expressed as a unit rate. CONSTANT RATE OF CHANGE- rate of change in a linear relationship.

  24. Constant Rate of Change The table shows the amount of money a booster club makes washing cars for a fundraiser. Use the information to find the constant rate of change in dollars per car. The table shows the number of miles a plane traveled while in flight. Use the information to find the constant rate of change in miles per minute. About 10 miles per minute

  25. Using a Graph to find Rate of Change The graph represents the distance traveled while driving on a highway. Use the graph to find the constant rate of change in MILES per HOUR. To do this, find any two points on the line such as (1,60) and (2,120) Use the graph to find the constant rate of change in miles per hour while driving in the city. 30 mile per hour Self Assessment: Complete p. 393 # 1-7 all by yourself. Then check your work with a partner.

  26. Slope7-2-C In a LINEAR RELATION, the vertical change per unit of horizontal change is always the SAME We call this change the SLOPE! Remember CONSTANT RATE OF CHANGE??? Well…it is the same as the slope. SLOPE basically tells how steep a line is.

  27. Find the SLOPE The table shows the relationship between the number of seconds y it takes to hear thunder after a lightening strike and the miles x you are from the lightening.

  28. Find the Slope Graph the data about plant height for a science fair project. Then find the slope of the line. EXPLAIN what the slope represents. The table below shows the relationship between the number of hours Carl works and the amount of money that he earns. Graph the data. Find the slope of the line! Then EXPLAIN what the slope represents.

  29. Interpret Slope Renaldo opened a savings account with the $300 he earned mowing yards over the summer. Each week he withdraws $20 for expenses. Draw a graph of the account balance versus time. Find the numerical value of the slope and interpret it in words. So, in order to draw the graph, you’re going to need a table. The x values should be the ______ and the y values should be the ______. The slope is $20/week. It is actually negative though b/c his balance declines! Self Assessment: Complete p. 398 # 1-4 all by yourself. Then check your work with a partner.

  30. Direct Variation7-3-C A car travels 130 miles in 2 hours, 195 miles in 3 hours, and 260 miles in 4 hours, as shown. What is the constant rate of change, or slope, of the line? Is the distance traveled always proportional to the driving time? Compare the constant rate of change to the constant ratio. DIRECT VARIATION: the relationship when two variables have a constant ratio Also called DIRECT PROPORTION The constant ratio itself is called CONSTANT OF VARIATION

  31. Find a Constant Ratio The height of the water as a pool is being filled is shown in the graph. Determine the rate in inches per minute. The rate of change is going to be constant because the graph forms a line. The pool fills at a rate of 0.4 inches every minute. EXAMPLE: Two minutes after a diver enters the water, he has descended 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the scuba diver descending?

  32. Direct Variation In DIRECT VARIATION, the CONSTANT RATE OF CHANGE, aka the slope, is assigned a special variable, k.

  33. Determine Direct Variation Pizzas cost $8 each plus a $3 delivery charge. Make a table AND graph to show the cost of 1, 2, 3, and 4 pizzas. Is there a constant rate? A direct variation? Because there is NO CONSTANT RATE, there is NO DIRECT VARIATION

  34. Determine Direct Variation Two pounds of cheese cost $8.40. Make a table and graph to show the cost of 1, 2, 3, and 4 pounds of cheese. Is there a constant rate? Is there a direct variation? Yes; Yes A photographer charges a $30 sitting fee and then $6 for each photograph ordered. Make a table and a graph to show the cost of 1, 2, 3, and 4 photographs. Is there a constant rate? A direct variation? INTERESTING AND IMPORTANT FACT: Just because there is a line/a linear relationship, there is not necessarily a direct variation. Yes; No

  35. Determine Direct Variation Determine whether each linear function is a direct variation. If so, state the constant of variation.

  36. In Summary… Self Assessment: Complete p. 408 # 1-5 all by yourself. Then check your work with a partner.

  37. Inverse Variation7-3-E What is the constant in each rectangle? What happens to the width as the length increases? What happens to the length as the width increases? This example shows INVERSE VARIATION INVERSE VARIATION means the product of x and y is a constant. We say that y is inversely proportional to x. As x increases, the value of y decreases, but not at a constant rate so the graph of an inverse variation is not a straight line.

  38. Inverse Variation The table shows the relationship between the frequency and wavelength of a musical tone. Determine if the relationship is an inverse variation. Justify your response. Look at the data. Graph the data in the table Analyze the graph. The graph shows that this is an INVERSE VARIATION. The x- and y- coordinates each have a product of 880.

  39. Inverse Variation Here’s how to solve: Use INVERSE VARIATION! Let x be the number of carpenters. Let y be the number of days. The number of carpenters needed to frame a house varies inversely as the number of days needed to complete the project. Suppose 5 carpenters can frame a certain house in 16 days. How many days will it take 8 carpenters to frame the house? Assume that they all work at the same rate. A crew of 8 carpenters can frame the house in 10 days.

  40. Inverse Variation Graph the data in the table. Then determine if the relationship is an inverse variation. The table shows the relationship between the cost of a gift and the number of friends splitting the cost. Determine if the relationship is an inverse variation. Justify your response. Graph the data in the table.

  41. Inverse Variation The number of hours it takes to wash windows at an office building varies inversely as the number of washers. Suppose two washers can get all of them washed in 64 hours. How many hours will it take 8 washers? Assume they all work at the same rate. The number of bricklayers needed to build a brick wall varies inversely as the number of hours needed. Four brick layers can build a brick wall in 30 hours. How long would it take 5 bricklayers to build a wall? 16 hours 24 hours Self Assessment: Complete p. 414 # 1-2 all by yourself. Then check your work with a partner.

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