2.2 Contextualizing Linear Functions

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1 2.2 izing Linear Functions Last unit we defined a linear function in several different ways: A function whose graph is a straight line, A function whose rate of change is constant, or A function whose equation is of the form where represents the slope and represents the y intercept. We ll now begin exploring the difference aspects of a linear function and how each piece affects the function as a whole. Input (Independent Variable) and Output (Dependent Variable) Since a function is a rule that assigns to each input exactly one output, it is crucial to identify and understand what the input and output are in the multiple forms of a function. Let s look at the following linear functions written in different forms. Example 1 The number of grapes depends on the number of branches off the main vine and is represented by this equation: Example 2 A dairy farmer can produce 25 gallons of milk from 3 cows Example 3 The cost for joining a gym includes a start up fee plus dues every month In example 1, what is the input? What do we start with? What comes first? The branches of the vine. Once we have branches, those branches then produce grapes. So the input, the thing we start with, is the independent variable. Cost in $ Months The independent variable is the variable that could be anything (at least anything within the domain). We could have any number of branches we want and the branches produce, or output, the grapes. That means that is the dependent variable. The number of grapes depends on the number of branches. In example 2, what is the input and what is the output? A farmer puts cows in his barn and gets out milk. The cows are the input meaning that is the independent variable. The milk is the output meaning that is the dependent variable. In example 3, what is the input and what is the output? What do we really want to know? The final cost. However, to find the cost we first have to know how many months you are going to be a member. That means that 66

2 the number of months is the input, or independent variable. Once we input the number months into the rule (which happens to be times 15 and then plus 20), we output the cost, which is the dependent variable. In the standard equation form of a linear function,, what is the input and output? Since we re talking about and on the coordinate plane, those are my input and output, but which is which? Generally speaking, but not always, the output is the variable by itself in any equation. In particular, in our generic form linear function, the variable is the output. That makes the input. If we plug in (input) an value then we get out (output) a value. Slope or Let s look at our examples again. Example 1 The number of grapes depends on the number of branches off the main vine and is represented by this equation: Example 2 A dairy farmer can produce 25 gallons of milk from 3 cows Example 3 The cost for joining a gym includes a start up fee plus dues every month In example one, the rate of change is given in the equation as the slope, or in the equation. Note that the slope is 100 which means that the rate of change is also 100, or in fraction form. This means that 100 grapes grow for every one branch. Months Perhaps it is easiest to see the rate of change (or slope) in the second example. How does the amount of milk change for the farmer? He gets 25 more gallons of milk for every 3 more cows he has, so we would write that rate of change as. In the third example, we ll need to find the slope by counting the rise and run. This is easiest to do from the intercept, the point where the line crosses the axis. Notice that the line crosses the axis at 20. The next nice point is at 1, 30. To get to that point from the intercept, you have to go up ten and right one. That means the slope, or rate of change, is 10. This means that the gym charges 10 dollars for every one month of membership. Cost in $ 67

3 Intercept or Let s look at our examples one last time. Example 1 Example 2 Example 3 The number of grapes depends on the number of branches off the main vine and is represented by this equation: A dairy farmer can produce 25 gallons of milk from 3 cows The cost for joining a gym includes a start up fee plus dues every month In example one, the initial is given in the equation as the intercept, or value in. Note that the initial value is 4. This means that only 4 grapes grow off the main vine no matter how many branches come off the main vine. Months The second example may be confusing because it s hard to see an initial value. An initial value would mean the amount of milk that the farmer starts with. Well, without any cows he wouldn t have any milk, so the initial value is 0. That s why there is no other number mentioned. We previously found the intercept for the third example to be 20. That means that no matter how many months you pay for membership to the gym, there will always be an additional $20 fee to pay. The problem context describes it as a start up fee. So if you pay for 3 months membership, you ll pay the $20 fee on top of the price per month. If you pay for 85 months of membership, you ll still pay the same $20 fee on top of the price per month. Cost in $ The Equation of the Line Once you know all the pieces, it s simply a matter of putting them in the right order. Since every linear function can be written in the form of, put the equation in that form. Remember that is the dependent variable (output), is the independent variable (input), is the rate of change (slope), and is the initial value ( intercept). For Example 2, we know that gallons of milk depends on the cows, the rate of change is, and the initial value is 0. Therefore, the equation is:. For Example 3, we know that the cost depends on the number of months you sign up for membership at the gym, the rate of change is 10, and the initial value is 20. Therefore, the equation is:

4 Lesson 2.2 Identify the rate of change, initial value, independent variable, and dependent variable. Then describe what the rate of change and initial value mean in the context of each situation. Finally, write the equation of the linear function. 1. A 2.5 foot rocket s distance traveled in meters based on time in seconds is modeled by the following function: 5 2. : : Cost in $ 2. The cost for 6 people to travel in a taxi in New York based on the number of miles driven is shown by the following graph: : : Miles driven 3. Planet Wiener receives $2.25 for every hotdog sold. They spend $105 for 25 packages of hot dogs and 10 packages of buns. Think of the linear function that demonstrates the profit based on the number of hotdogs sold. : : 69

5 4. The weight (in pounds) of a 20 x 10 x 12 aquarium tank based on the number of gallons of water inside is modeled by the following function: : : Profit 5. The amount of profit of the lemonade stand on 120 W Main Street based on the number of glasses of lemonade sold is modeled by the following graph: : : # of glasses sold 6. A candle starts at a height of 5 inches and diameter of 3 inches and burns 1 inch every 2 hours. Think of the linear function that demonstrates the height of the candle in terms of the time it has been burning. : : 70

6 7. The cost to stay in a 4 star hotel each night is modeled by the following function: _ : : 8. The cost to attend a sports clinic 37 miles away based on the number of days attended is modeled by the following graph: _ Cost : : Days 9. A dog kennel charges $40 for each night the dog stays in the kennel. Each day includes a 2 hour play time and 1 hour etiquette training. The kennel also charges a $10 bathing fee for a bath before the dog returns home. Think of the linear function that demonstrates the cost of putting a dog in the kennel in terms of the number of nights. _ : : 71

7 10. The number of gallons of gas in your 15 gallon gas tank based on the number of miles traveled is modeled y the following function: 12. _ : : 11. The number of pizzas ordered for 8 th grade night based on the number of students is shown by the following graph: Number of pizzas : : Number of students 12. It costs $5.50 to mail a large package to New Zealand. The post office will weigh your package and charge you an extra $0.30 per pound. The delivery takes 2 weeks. Think of the linear function that demonstrates the cost to mail a large package to New Zealand based on the number pounds it weighs. : : 72

8 13. An author wrote an 876 page book. The amount of profit based on the number books sold is modeled by the following function: _ : : Grade earned 14. The average grade earned on the Unit 3 test based on the number of hours of studying is modeled by the following graph: _ : : EQ of Line: Hours of studying 15. Kiley invited 32 people to her 13 th birthday party at the bowling alley. She hopes most people can come! It costs $40 to reserve the bowling alley. It will cost an additional $2 per friend to bowl. Think of the linear function that demonstrates the cost of the birthday party in terms of the number of friends who attend and bowl. : : 73

9 16. You started a mowing business so you could buy a 2015 Chevy Camaro when you turn 16. The amount of money in your bank account based on the number of yards you mow is modeled by the following function: 30. : : Temperature 17. When an oven is set at 350, the internal temperature of a chicken breast after every minute it s in the oven is modeled by the following graph: _ : : EQ of Line: Minutes 18. Walter s Water Adventures charges $34 to enter. This fee helps pay for maintenance and lifeguards. They always have 3 lifeguards at each slide plus 2 watching the wave pool. Think of the linear function that demonstrates the number of lifeguards on duty based on the number of slides open on a given day. : : 74