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Week 6: Functions, Equations, and GraphsFunction: shows the relationship between values from one set of numbers to another set.Input: the first set of numbers. represented by “x” in an ordered pairis the independent variable Output: the second set of numbersrepresented by “y” in an ordered pairis the dependent variable (it will change according to the value of the input. )00Example: Identify the dependent and independent variables in this situation. Then write an equation to represent the relationship.Part A: Sam is trying to figure out his total cost for the bus rides he has taken while on vacation. Each bus ride costs $2.50. Rides: independent variable (input) Cost: dependent variable (output) Equation: cost = $2.50 x the number of rides c = 2.50rPart B: How much will it cost for 4 bus rides?C = 2.50(4)C = $10.00It will cost $10 for 4 bus rides. Example: Identify the dependent and independent variables in this situation. Then write an equation to represent the relationship.Part A: Sam is trying to figure out his total cost for the bus rides he has taken while on vacation. Each bus ride costs $2.50. Rides: independent variable (input) Cost: dependent variable (output) Equation: cost = $2.50 x the number of rides c = 2.50rPart B: How much will it cost for 4 bus rides?C = 2.50(4)C = $10.00It will cost $10 for 4 bus rides. You Try: Janna is buying a laptop and a flash drive. The total cost, c, will include the price, p, of the laptop, plus $12.50 for the flash drive.Independent Variable:Dependent Variable:Equation: __________________________________ Find the total cost if the price of the laptop is $375.00Kenneth earns $9 per hour mowing lawns. The total amount earned, t, will depend on the number of hours, h, he works. Which equation represents this situation?t = 9hh = 9tt = 9 + hh = 9 + tHakeem wants to rent a bike. Bike rental costs $1.25 per hour plus a one-time fee of $3. Which equation represents the relationship between the number of hours, h, Hakeem rents the bike and the total cost, c?h=1.25c + 3c = 1.25h + 3h = 3c + 1.25c = 3h + 1.25Neema purchases a gallon of milk for $3 and some tomatoes fro $2 per pound. Which equation represents the relationship between the number of pounds, p, of tomatoes purchased and the total amount, a, Neema spends.p = 3a + 2a = 3p + 2p = 2a +3a = 2p + 3Function Tables: 0179070Example: Bob’s school club washes cars to earn money. They earn $5 for each car they wash. Write an equation to represent the amount of money the club can earn.Equation: 5 × input = output5x = y let x represent the input, and y represent the output.Make a function table to show the amount of money Bob’s school earns washing 5,10, 15, and 20 cars. Input (x)Output (y)5251050157520100The rule is “multiply by 5”The input values are 5, 10, 15, 20Each output is 5 times the corresponding input. 0Example: Bob’s school club washes cars to earn money. They earn $5 for each car they wash. Write an equation to represent the amount of money the club can earn.Equation: 5 × input = output5x = y let x represent the input, and y represent the output.Make a function table to show the amount of money Bob’s school earns washing 5,10, 15, and 20 cars. Input (x)Output (y)5251050157520100The rule is “multiply by 5”The input values are 5, 10, 15, 20Each output is 5 times the corresponding input. You can also determine a rule and write an equation when given a function table:069850Example: Write an equation to show the relationship shown in this table:Input (x)3035404550Output (y)678910Find a pattern: Think “What can I do to each ‘x’ value to find its corresponding ‘y’ value?- Since the y values are less than the x values, you know it must be dividing or subtractingThe pattern each time is to divide x by 5 to get y.The equation is y = x ÷ 5 or y = x50Example: Write an equation to show the relationship shown in this table:Input (x)3035404550Output (y)678910Find a pattern: Think “What can I do to each ‘x’ value to find its corresponding ‘y’ value?- Since the y values are less than the x values, you know it must be dividing or subtractingThe pattern each time is to divide x by 5 to get y.The equation is y = x ÷ 5 or y = x5You Try:Write an equation to represent the relationship shown in the table. Then find the missing value in the table:X20406080Y2343831) Equation: ________________________________2) X3456Y182439Equation: ________________________________Use the equation to complete the table:y = 7x4) y = x – 2X581114YX1234YA bowling ally charges $1.50 for shoe rental and $2 per game. The table shows the total cost, y, of bowling x games:Number of Games, xCost, y13.502.5.503?49.50Which equation could be used to find the total cost, y of bowling x games?y = 2x + 1.5y = 1.5x + 2y = x + 2y = x + 1.5What is the total cost of bowling 3 games?$7.50$6.50$50.00$4.50The table below shows the price of student notebooks at a school store. Each notebook is the same price.Number of NotebooksPrice in $264126188249?According to the relationship shown in the table, what would be the price, in dollars, of 9 notebooks?25273042GraphsA straight line on a coordinate plane can represent a linear function. Remember:The horizontal axis is the x-axis (going across)The vertical axis is the y-axis (going up and down)Ordered pairs are always written in the form (x,y) (input, output) Example 1: Graph the function y = 3x on a coordinate plane.Step 1:Make a table of values for this function. You can choose any values for x, and then determine the corresponding y-value. It is usually best to pick x = 0,1,2,3…XY0013263925520654191000The values from this table can be written as the ordered pairs:(0,0), (1,3), (2,6), (3,6)Step 2: Draw a coordinate plane. Graph each point from the table on the plane. Then connect the points with a straight line going in each direction. **Remember, the points on the linear function can have positive and negative coordinates too. In this example, the point (-2,-6) is on the line because if x = -2, y = 3(-2) = -6** In some situations, it does not make sense to have negative values for x and y. In these cases, the graphs should appear only in quadrant one. (Example, how far a car travels, cost for an item…)Example 2: Graph the function y = 4x + 1Step 1: Make a table of values377190026162000For this function, I will pick some negative values for x. I want to pick smaller numbers, because we are going to multiply by 4 and then add 1. This will keep the y value lower and easier to graph. XY-2-7-1-30115 Step 2: Graph the ordered pairs from the table. Connect the points with a straight line.(-2,-7), (-1,-3), (0,1), (1,5) 2286000-11430000You Try:Draw the graph of :y = x – 2 .xyDetermine the relationship between x and y from the table below. Write an equation to represent this relationship. Then graph the equation on the coordinate plane:34290009588500xy2131.54252.563 Equation: y = More Practice: There are 12 boys in a math class. The total number of students s depends on the number of girls in the class g. Which equation represents this situation?g = 12ss = 12gg = 12 + ss = 12 + gA store received a shipment of soup cans. The clerk put an equal number of cans on each of 4 shelves. Which equation represents the relationship between the total number of cans t and the number of cans on each shelf n?n = t ÷ 4n = t – 4n = 4tn = t + 4Marcus sold brownies at a bake sale. He sold d dollars worth of brownies. He spent $5.50 on materials, so his total profit p can be found by subtracting $5.50 from his earnings. Which equation represents this situation?p = d + 5.50p = d – 5.50p = 5.50dp = d ÷ 5.50A balloon rises at the rate of 12 feet, f, per second, s. Which equation represents this scenario?f = 12sf = 12 + sf = 12 - sf = s/12Which table represents the scenario in the previous problem?f2468s12243648 s1234f12243648 s12243648f1234s1234f12362448Which table represents the equation t = 20d?a. c. t2468d20406080b. d2468t20406080 t1234d20406080d. d1234t20406080 It takes Phillip 4.5 minutes to run a mile. Let m represent the number of minutes and d represent the number miles ran. What is the missing value in the table?m4.5918d1234313.51317.5Which equation represents the function shown in the table?x8101214y46810y = 2xy = x/2y = x - 4y = x +4391160013335 Which scenario can the graph represent?Melissa earns $12 per hourThe temperature increased by 2 per hourZac ran a mile every 15 minutesMaria folds 10 napkin every minute Mel wants to graph the function represented by the table. Which ordered pair is a point on the graph of the function?x1234y8162432(1,2)(3,24)(8,16)(32,4) A bookstore charges $4 for all children’s books. Michael graphs the function that gives the total cost, y, in dollars to buy x number of books. Which ordered pair is a point on the graph of the function?(2,8)(5,2)(4,12)(9,18) If x < c and x = 17, which of the following could be values of c?A. (15, 16, 17)B. (17, 18, 19) C. (13, 20, 22)D. (19, 24, 27)Darryl has 5 hotdogs at home. He goes to the store and purchases more. Each pack has 8 hot dogs. Which equation displays the number of hot dogs Darryl has? Explain.A. y = 5x + 8 B. y = 5x – 8 C. y = 8x + 5D. y = 8x – 5 Which equation describes the table?x 1 2 3 4 y 1.23 2.46 3.69 4.92A. y = 2xC. y = x + 1.23B. y = 1.23xD. y = x/1.23Which equation describes the table?x y 0 5 2 9 3 11 7 19 A. y = 5xB. y = 5x + 2C. y = 2x + 5D. x = 3y + 5Feet 3 6 9 15 24 Yards 1 2 3 5 ?What number should replace the question mark?A. 6B. 7C. 8D. 9E. 12Miles works at the Apple Store for 6 hours and makes $12.25 per hour plus a sales bonus of $18.65. How much did he make in total? A. y = $92.15B. y = $86.35C. y = $78.75D. y = $96.35Which statement is true?A. x = output, y = inputB. x = constant, y = coefficientC. x = input, y = outputD. x = coefficient, y = constantThe graph at the right shows the cost of buying comic books from an online retailer. The shipping charge is always the same no matter how many comic books are purchased. 45720001276350Use the graph.Fill in this table of values using the coordinates on the graph:XY01234537210060960(2,8)00(2,8) How much does each book cost?$1b. $25029200160655(1,5)00(1,5)c. $3d. $5What is the shipping cost? $1b. $3c. $2d. $5 Which equation represents the total cost in dollars c for n comic books?c = 3n + 2b. c = 3nc. c = 2n + 3d. c = 2nThe linear equation y=5x represents the cost y in dollars of x pounds of dog food. Which ordered pair lies on the graph of this equation?(3,15)(3,10)(2,3)(2,12)Dee is driving at an average speed of 50 miles per hour. Write an equation for the relationship that gives the distance y, in miles that Dee drives in x hoursEquation: _____________________________How many miles will Dee drive in 4 hours?How many miles will Dee drive in 3? hours?Simplify. Follow the Order of Operations.54 ÷ 9 ? 3 ? 20b. 1c. 3d. 610 + 32 ? 2 – 830 b. 24c. 20d. 14 Evaluate p + (n ? 1) ? 3 for n = 7 and p = 95145c. 31d. 27 Evaluate 18 ? z ? (i2 ?8) for i = 4 and z = 20 218128 Evaluate e2 + 4w ÷ 2 ? 1 for e = 6 and w = 3.41 232017 ................
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