Topic: Linear Functions Essential Question: What everyday relationships can we represent using graphs, tables and functions? Learning Goal(s) Students

Topic: Linear Functions

Essential Question: What everyday relationships can we represent using graphs, tables and functions?

Learning Goal(s)Students will be able to:□ Construct a function to model a linear relationship between two quantities□ Determine the rate of change and initial value of a linear function□ Interpret the rate of change and initial value of a linear function

Question(s)/Concepts NOTES & EXAMPLES You HAVE 8 MINUTES TO COMPLETE.

TO DO NOW: 1) Use the given values to make a table of solutions: y = 2x – 1 when x = 1, 2, 3, 4 2) Begin filling out the “K” and “W” on your K-W-L once you finish task #1.

Correct Response for #1X 2x - 1 Y (X, Y)1 2(1) – 1 1 (1,1)

2 2(2) – 1 3 (2,3)

3 2(3) – 1 5 (3,5)

4 2(4) – 1 7 (4,7)

Intro to Functions

Question(s)/Concept(s) NOTES & EXAMPLES

PROCESSING Our school in the City: New York, New YorkIf we can convince our principal, using functions, equations, graphs, and tables that if we have certain fund-raisers, particularly a car wash, we may be able to take a trip to New York!

So let’s talk business so that we can prepare a presentation with CRAZY SWAG so that our principal simply cannot refuse. 1st – Let us understand the LANGUAGE we need to use in the presentation.

OK, so if we are going to have a car wash our focus will be the money we are going to make from it.


PROCESSING Our school in the City: New York, New York

Cars (X-VALUES) Money (Y-VALUES)

1 3

3 9

5 15

7 21

INPUT (Independent

variable)

OUTPUT ($$S)(Dependent

Variable)

OrderedPair

a) Look at the table to determine how much are we charging to wash each car? $3. Is that the rate of change? Yesb) What is the general rule or equation that we can use to figure out how much ANY # of cars washed cost? y = 3x

Temperature Check



Cars (X-VALUES a.k.a Domain)

Money (Y-VALUES a.k.a Range)

1 3

3 9

5 15

7 21

INPUT (Independent

variable)


Variable)

OrderedPair

The cars washed and the money earned is what we call a RELATION. A RELATION is a set of ordered pairs. They have a special type of relation which is known as FUNCTION. In a function, like this one, each input or DOMAIN VALUE is paired with EXACTLY ONE output or RANGE VALUE. What is the rate of change?

Temperature Check





1 3

3 9

5 15

7 21

INPUT (Independent

variable)


Variable)

OrderedPair

Now that we know what the general rule is for determining the money earned for washing any number of cars, figure out how much would it cost to wash 4 cars? f(x) = 3(4), f(x) = $12

Our Rule is:f(x)=3x

Temperature Check





1 3

3 9

5 15

7 21

INPUT (Independent

variable)


Variable)

OrderedPair

So we heard through the grape vine that a celebrity is considering sponsoring our trip. The celebrity decides to give us $1000 towards the trip. What will the new function rule be? f(x) = 3x + 1000.





1 3

3 9

5 15

7 21

INPUT (Independent

variable)


Variable)

OrderedPair

The new rule will be f(x) = 3x + 1000. What does the $1,000 represent? The money that the students already have because of the celebrity donation.

Temperature Check

14

Q’s & C’s NOTES & EXAMPLES

Using Slopes and Intercepts

We have already learned that the rate of change for a linear function is constant and is known as the slope of a line represented by the letter m, is a ratio that compares the difference in y-values (independent variables) to the difference in x-values (dependent variables) for any 2 points on the line.

In our fundraiser, we were fortunate enough to have a celebrity offer to donate a lot of money towards our trip to New York City. When we think about our function rule now, what do we need to take into consideration and why?

1000

15

Q’s & C’s NOTES & EXAMPLES

Using Slopes and Intercepts

We have already learned that the rate of change for a linear function is constant and is known as the slope of a line represented by the letter m, is a ratio that compares the difference in x-values (independent variables) to the difference in y-values (dependent variables) for any 2 points on the line.

In our fundraiser, we were fortunate enough to have a celebrity offer to donate a lot of money towards our trip to New York City. When we think about our function rule now, what do we need to take into consideration and why? Y-INTERCEPT:where the line crossesthe Y-AXIS. This point has x-value of 0 and ONLY one y-value.

1000

Initial Value or

Starting Point





1

3

5

7

INPUT (Independent

variable)


Variable)

OrderedPair

We need to construct a new table of values to graph the new function rule or equation y=3x+1000. The x-values are given in the table already. Complete the table by determining the range values.





1 1003

3 1009

5 1015

7 1021

INPUT (Independent

variable)


Variable)

OrderedPair

We need to construct a new table of values to graph the new function rule or equation f(x)=3x+1000. The x-values are given in the table already. Complete the table by determining the range values.

Temperature Check

Quick Checks1) Use the table below to find the value of y when x = 5

X Y

2 9

4 19

6 29

8 39

2) Lorenzo needs to construct a table of values to graph the equation y = -2x + 1. The x-values are given below. Which of the following is the correct list of y-values for this table?

X Y -2 5

-1 3

0 1

1 -1

2 -3

A) 2, 3, 7, 1, 4B) -5, -3, 0, -1, -3C) -5, -3, 0, 1, 3D) 5, 3, 1, -1, -3

Quick Checks3) At the local movie theater it cost $12.00 for 2 students to see a movie. It costs $18 for 3 students, and it cost $24.00 for 4 students. Since the students are going as a group, the group will pay one fee of $30 for each student to get a snack and drink. 1. What is the function rule that relates the number of students

to the total cost the group would have to pay, when they go to the local movie theater? x= the number of students; f(x)=6x+30

2. What is the initial value? 303. What is the rate of change? What does the rate of change

represent? 6, the ticket cost for one student

Trivia Question

WORTH 5 EXTRA CREDIT POINTS

Is it true that you have to go either up or down the y-axis first when graphing? No b/c we need to move along the x-axis first.

Exit Ticket#1 - At the local movie theater it cost $16.00 for 2 students to see a movie. It costs $24 for 3 students, and it cost $32.00 for 4 students. Students must also pay $3.00 for a wrist band which allows them to buy food. What is the function rule that relates the number of students to the total cost they would have to pay when they go to the local movie theater? A) f(x) = 3x + 8B) f(x) = 3xC) f(x) = 8x - 3D) f(x) = 8x + 3 correct response

#2 – Use the table below to determine the value of y when x = 8

A)21 B) 2 correct response C) 8 D) none of above

#3- A student needs to construct a table of values to graph the equation y = -3x + 2. Which of the following is the correct list of y-values for this table given the following x-values: 1, 2, 3, 4? A)-1, -4, -7, -10 correct response C) 1, 2, 3, 4B) 1, 4, 7, 10 D) 5, 8, 11, 14

X Y

2 8

4 13

6 18

Practice Makes Perfect

All students need to work on the following problems assigned to your group IN CLASS NOW (To be checked). YOU HAVE 10 minutes to complete this activity.

HOMEWORK HW ASSIGNED Due:

Groups

Assignment Color

Powerhouse 1) Complete the worksheet handed out for your group. It is titled, “Practice B”

Jr. Powerhouse 2) Complete the worksheet handed out for your group. It is titled, “Practice A”

Powerhouse In-Training

3) Complete the worksheet handed out for your group. It is titled, “Review for Mastery.”

Independent Practice AnswersWord K W L

Relation

Domain

Range

Function

Independent Variable

Dependent Variable

Vertical Line Test

Retain for Mastery In your section titled, “Summary,” respond to the following questions, 1) How can we use functions to represent relationships?

Documents

Topic: Linear Functions Essential Question: What everyday relationships can we represent using graphs, tables and functions? Learning Goal(s) Students