Activity 2 - 11

Long Distance by Phone

Objectives

• Graph a piecewise linear function

• Write a piecewise linear function to represent a given situation

• Determine if a piecewise linear function is continuous

• Graph a function defined by y = |x - c|

Vocabulary• Piecewise function – a function that is defined by more

than one equation over its domain

• Absolute value – symbols: | |; gives only the positive result of the function

Activity

A certain lone-distance telephone carrier offers the following rates for calls outside the state in which you live: $0.15 per minute for the first 10 minutes, $0.08 per minute for each minute thereafter. Also, whereas most companies round up to the next minute for any fraction of a minute, this company charges for the exact duration of your call.

Complete the following table:

T (in min) 1 2 5 10 15 20 30

C (in $) .15 .30 .75 1.50 1.90 2.30 2.70

Activity cont

Write an equation for the cost for calls

Where does the “break point” occur in the cost?

What is the cost of a 6-minute call?

Determine the cost of a 33-minute call and interpret the result

Break point at 10 minutes

0.15 m 0 < m ≤ 10C = 0.08 m 10 < m

(0.15) (6) = 0.90

First 10 minutes + other 23 minutes (0.15) (10) + (0.08) (23) = 1.50 + 1.84 = 3.34

Activity cont

Sketch the graph of the cost function

Now use your calculator to graph it. Refer to instructions on page 263

y

x

Competitor Offer

A competitor offers $0.99 for the first 20 minutes and $0.07 for each minute thereafter.Complete the following table:

Write an equation that gives the cost for calls less than 20 minutes

Write an equation that gives the cost for call more than 20 minutes

T (in min) 1 5 10 20 25 30

C (in $) .99 .99 .99 .99 1.34 1.69

Cost = 0.99 0 < m ≤ 20

Cost = 0.99 + 0.07(M – 20) 20 < m Cost = 0.07M – 0.41

Competitor Offer cont

A competitor offers $0.99 for the first 20 minutes and $0.07 for each minute thereafter.

Write the piece-wise function for the cost of this long-distance service

Determine the cost of a 33 minute call

0.99 0 < m ≤ 20C = 0.07 m – 0.41 20 < m

Cost = 0.99 + 0.07(13) = 0.99 + 0.91 = 1.90 Cost = 0.07M – 0.41 = 0.07(33) – 0.41 = 2.31 – 0.41 = 1.90

Cost of Natural Gas

Your natural gas bill is computed base on units of 1000 cubic feet. There are two parts to the bill: the gas charges and the delivery charges. The most expensive part is the delivery charges. For residential customers, the gas delivery charge for natural gas is billed in the following manner:

$40.74 for 0 to 3000 cubic feet$0.98 per thousand feet for 3000 to 50000 cubic feet$0.15 per thousand feet for each thousand cubic feet in excess of 50000•Complete the following table:

x, K cu ft 0 1 2 5 10 50 100 200

C, Monthly cost (in $)

Cost of Natural Gas cont• Write an equation for the delivery charge, if you use

3000 cubic feet of less

• Write an equation to represent the delivery charge if the number of thousand cubic feet used is greater than 3 and less than 50. Remember x represents the total number of thousand cubic feet used.

• Write an equation to represent the delivery charge if the number of thousand cubic feet used is more than 50.

Cost of Natural Gas cont 2

• Write the piece-wise function for the delivery charge of gas

• What is the delivery charge if you use 157 thousand cubic feet of gas?

• If it costs $1.721 for each thousand cubic feet of gas used (supply charge), then what was your total gas bill that month?

Absolute Value

• What is the domain of the function f = |x|?

• Complete the following table:

• Describe the shape of the graph

• What is the range of the function?

x -4 -3 -2 -1 0 1 2 3 4

f(x)

Domain x value x (x is any real #)

Looks like a ‘V’

Range y-values y 0

Absolute Value• Sketch a graph of the function f = |x|.

• Describe the shape of the graph

• Sketch it on your calculator

• What is the range of the function f(x) = |x – 2|?

y

x

Looks like a V

Range y y 0

Y1 = abs(x) get abs( from catalog (2nd 0)

Absolute Value cont• Sketch a graph of y = |x – 2|.

• What type of shift occurred?

• Sketch a graph of y = |x| + 2

• What type of shift occurred?

y

x

Shift right of 2

Shift up of 2

Summary and Homework• Summary

– Piecewise function is a function that is defined differently for certain “pieces” of its domain

– The absolute value is a special piecewise function:

– The absolute value of a linear function, g(x) = |x – c|, always has a v-shaped graph with a vertex at (c, 0)

• Homework– Pg 267–71; 1-6

x if x 0|x| = -x if x < 0