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

GOAL 1 INPUT-OUTPUT TABLES EXAMPLE Tables and Graphs 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.

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. a.Make an input-output table. b. Is this a function? c. Describe the domain and range. EXAMPLE 2 n …300 p n …300 p04812…920 Yes; none of the inputs are repeated. Domain: 70, 71, 72, 73,…, 300 Range: 0, 4, 8, 12,…,920

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. a.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. Extra Example 2 t d t d

Extra Example 2 (cont.) t d

Input-Output Table Description in Words Equation Graph 4 WAYS TO DESCRIBE A FUNCTION 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.

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 = t for 0 t 4. 1.Make a table (use 0, 1, 2, 3, and 4 minutes). 2.Draw a line graph. 3.Describe the domain and range.

Checkpoint (cont.) t01234 d

Checkpoint (cont.) Domain: all numbers between and including 0 and 4 Range: all numbers between and including 2000 and 6000 t01234 d 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 ft are included as the plane climbs.

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

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. a.Write an equation. b.Create an input-output table for hours c.Make a line graph.

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

Extra Example 3 (cont.) h c

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. 1.Write an equation. 2.Make an input-output table, using a one-half hour interval. 3.Make a line graph.

Checkpoint (cont.) a. T = h b. h T c.

QUESTIONS?