MPM1DI – Unit 5: Linear Relations– Lesson 1 Date:___________ Learning goal: how to determine the type of variation, initial value, rate of change, and equation given a variety of information. Initial Value and Rate of Change WARM UP Nisha is just learning how to snowboard. White Mountain charges $10/hour for lessons and $40 for the lift ticket and snowboard rental. a) Write an equation to model the relationship. Be sure to declare your variables. b) Make a table of values for up to 5 hours, then graph the relationship c) Use your equation to determine how much it will cost in total for Nisha to take 2.5 hours of lessons? d) If Nisha paid $75, Use your equation to determine how long she was at the White Mountain getting lessons? e) This relationship is called a linear relationship. Name the 4 ways to represent a linear relationship. 0 1 2 3 4 5
Transcript
MPM1DI – Unit 5: Linear Relations– Lesson 1 Date:___________ Learning goal: how to determine the type of variation, initial value, rate of change, and equation given a variety of information.
Initial Value and Rate of Change WARM UP Nisha is just learning how to snowboard. White Mountain charges $10/hour for lessons and $40 for the lift ticket and snowboard rental. a) Write an equation to model the relationship. Be sure to declare your variables. b) Make a table of values for up to 5 hours, then graph the relationship c) Use your equation to determine how much it will cost in total for Nisha to take 2.5 hours of lessons? d) If Nisha paid $75, Use your equation to determine how long she was at the White Mountain getting
lessons? e) This relationship is called a linear relationship. Name the 4 ways to represent a linear relationship.
0
1
2
3
4
5
0 100 200 300 400 500 600 700 800 900
1000
0 1 2 3 4 5 6 7 8 9 10
Cos
t ($)
# of Days Rented
Cost of Renting a Car
INITIAL VALUE AND RATE OF CHANGE Example 1: Henry is considering renting a car. Analyze the following graph, then answer the questions below. a) What is the independent variable? b) What is the dependent variable? Explain how you know. Explain how you know. b) What is the cost if he cancels his reservation? c) What is the cost of renting the car per day? d) Describe the cost of renting the car in words. e) Describe the cost of renting the car using an equation. Be sure to declare your variables.
Initial Value • This is where the line crosses the _____________________________________________ • It is the known as the ________________________________________________________
Rate of Change
• This is how the dependent variable changes with respect to the independent variable
• It is the rate per unit f) Create the table of values for the cost of renting a car.
g) Where does the initial value appear the table of values? h) Where does the rate of change appear the table of values? g) Which form of a linear relation is it easiest to determine the … Initial value? Rate of change?
Rate = ________________________
0
2
4
6
8
10
Example 2: Andrew is considering renting a canoe. He has found two companies to choose from. Analyze the initial value and rate of change in each relationship.
Option #1 -‐ Big Pine Outfitters
Table of Values Graph
Option #2 – Hemlock Bluff
Table of Values Graph
Time (h) Cost ($)
0 40
2 60
4 80
Time (h) Cost ($)
0 0
1 30
2 60
3 90
4 120
Equation C = 10h + 40
Description Big Pine Outfitters charges a base fee of $40 to deliver the canoe to the launch site and $10 per hour of use.
Equation C = 30h
Description Hemlock Bluff charges $30 per hour.
TYPES OF VARIATION Example 3: For each of the following, determine whether they are examples of direct or partial variation. Direct / Partial Direct / Partial Direct / Partial Direct / Partial Direct / Partial Direct / Partial
Direct / Partial Direct / Partial Direct / Partial Direct / Partial Direct / Partial Direct / Partial Determine the initial value and rate of change for each of the relationships above.
0 20 40 60 80
100 120 140
0 2 4 6 8 10
Mon
ey Earne
d ($)
Hours Worked
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10
Cost($)
Hours Worked
0
500
1000
1500
2000
2500
0 2 4 6 8 10
Mon
ey Owed
($)
Months
Direct Variation
Partial Variation
A painter charges $50 per hour.
A moving company charges a fee of $100 plus $75 per hour.
A banquet hall charges a $700 booking fee and
$30 per person.
C = 6n + 100
C = 25n C = 70 – 10n
Assignment 5.1: Initial Value and Rate of Change For questions 1-‐7, answer the following questions below.
a. Indicate if the relation is a partial or direct variation, b. Identify the initial value and rate c. Write the equation for each relationship in the space provided. d. Be sure to declare your variables any calculations that you have made.
1. A family meal deal at Chicken Deluxe costs $26, plus $1.50 for every extra piece of chicken added to the
bucket. 2. A Chinese food restaurant has a special price for groups, dinner for two costs $14 per meal 3.
Number of Toppings
Cost of a Large Pizza ($)
0 9.40 1 11.50 2 13.60 3 15.70 4 17.80
4.
Number of Scoops Cost of Ice Cream with
Sugar Cone ($) 0 1.25 1 2.00 2 2.75 3 3.50 4 4.25
5.
6.
7.
Age of Vehicle (years)
Depreciated Value of a Mid-‐size Car
Value ($)
1 2 3 4 5
30000 25000 20000 15000 10000 5000
Number of Guests
Cost of Holding a Formal at a banquet Hall
Cost ($)
20 40 60 80 100
1200 1000 800 600 400 200
Number of Hot Dogs
Total Cost of Hot Dogs at the Baseball Game
Cost ($)
1 2 3 4 5 6
24 20 16 12 8 4
8. This scatter plot shows the relationship between the rebound height of a ball and the height from which the ball is dropped. Draw a line of best fit for the data on the grid above. Determine an equation for your line of best fit.
9. The Radicals, a small high school band, recently signed a
contract with a record label. Their earnings include a signing bonus plus an amount per CD sold, as shown in the table below.
a) Determine the amount of the signing bonus and the amount they receive per CD.
b) Determine an equation to model this situation. 5.1 Answers 1. C = 1.50p + 26 2. C = 14m 3. C = 2.10t + 9.40 4. C = 0.75s + 1.25 5. V = -‐2500y + 30000 6. C = 10g + 100 7. C = 2h 8. Answers may vary à R = 0.5d 9. a) Signing bonus is $10 000, amount per CD is $0.12/cd, b) E = 0.12c + 10 000
