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1.7

1.7. An Introduction to Functions. 1. 2. GOAL. GOAL. Identify a function and make an input-output table for a function. Write an equation for a real-life function, such as the relationship between water pressure and depth.

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1.7

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  1. 1.7 An Introduction to Functions 1 2 GOAL GOAL Identify a function and make an input-output table for a function. Write an equation for a real-life function, such as the relationship between water pressure and depth. To represent real-life relationships between two quantities such as time and altitude for a rising hot-air balloon. Whatyou should learn Why you should learn it

  2. 1.6 Tables and Graphs INPUT-OUTPUT TABLES 1 GOAL EXAMPLE 1 • VOCABULARY • function • input/domain • output/range • input-output table In a function, each input has exactly one output. Another way to put it is no number in the input can be repeated.

  3. EXAMPLE 2 Extra Example 1 • The profit on the school play is $4 per ticket minus $280, the expense to build the set. There are 300 seats in the theater. The profit for n tickets sold is • p = 4n – 280 for 70 ≤ n ≤ 300. • Make an input-output table. • b. Is this a function? • c. Describe the domain and range. Yes; none of the inputs are repeated. Domain: 70, 71, 72, 73,… , 300 Range: 0, 4, 8, 12,… ,920

  4. Extra Example 2 • You bicycle 4 mi and decide to ride for 2.5 more hours at 6 mi/hr. The distance you have traveled d after t hours is given by d = 4 + 6t, where 0 ≤ t ≤ 2.5. • Make an input-output table. Calculate d for each half-hour (t = 0, 0.5, 1, 1.5, 2, 2.5). • b. Draw a line graph.

  5. Extra Example 2 (cont.)

  6. 4 WAYS TO DESCRIBE A FUNCTION • Input-Output Table • Description in Words • Equation • Graph By the end of the lesson you should be able to move comfortably among all four representations. You will then have a variety of ways to model real-life situations.

  7. Checkpoint • A plane is at 2000 ft. It climbs at a rate of 1000 ft/min for 4 min. The altitude h for t minutes is given by • h = 2000 + 1000t for 0 ≤ t ≤ 4. • Make a table (use 0, 1, 2, 3, and 4 minutes). • Draw a line graph. • Describe the domain and range.

  8. Checkpoint (cont.)

  9. Checkpoint (cont.) Domain: all numbers between and including 0 and 4 Range: all numbers between and including 2000 and 6000 All numbers are included because time is continuous. This is what is shown by connecting the data points with a line. Even numbers such as 1.73 minutes or 2148.4 ft are included as the plane climbs.

  10. 1.7 An Introduction to Functions 2 WRITING EQUATIONS FOR FUNCTIONS GOAL EXAMPLE 3 • Use the problem solving strategy from Section 1.5 to: • Write a verbal model • Assign labels • Write an algebraic model

  11. Extra Example 3 • An internet service provider charges $9.00 for the first 10 hours and $0.95 per hour for any hours above 10 hours. Represent the cost c as a function of the number of hours (over 10) h. • Write an equation. • Create an input-output table for hours 10-14. • Make a line graph.

  12. Extra Example 3 (cont.) Rate per hour VERBAL MODEL Connection fee Number of hours Cost = + • LABELS c $9 $0.95 h ALGEBRAIC MODEL c = $9 + $0.95h

  13. Extra Example 3 (cont.)

  14. Checkpoint • The temperature at 6:00 a.m. was 62°F and rose 3°F every hour until 9:00 a.m. Represent the temperature T as a function of the number of hours h after 6:00 a.m. • Write an equation. • Make an input-output table, using a one-half hour interval. • Make a line graph.

  15. Checkpoint (cont.) a. T = 62 + 3h b. c.

  16. QUESTIONS?

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