Functions (book 2.1 p. 61)

Objective: I can determine when a relation is a function, and evaluate functions using function notation.

Relation: A set of ordered pairs

Ex.

Domain:

Range:

Ex. Given the following relation: State the domain and range.

{(1, 2); (-3, 4); (-3, 1); (5, 6)}

Function:

Example 1: Determine the domain and range of each relation. Then determine if the relation is a function.

x

y

2

-2

-1

-1

-2

0

-1

1

2

2

a) {(-6, -1), (-5, -9), (-3, -7),(-1, 7), (6, -9)}b) c)

Vertical Line Test: When given a graph, no vertical line can intersect the graph in more than one point for it to be a function. If a vertical line hits the graph at more than one point it is NOT a function.

Example 2: Determine which of the graphs below represents a function.

Function Notation:

y = 2x2 - 8 f(x) = 2x2 - 8

Example 3: Evaluate the function for each value given.

f(x) = 2x2 - 8

a) f(6)b) f(-2)

Example 4: Evaluate the function for each value given.

g(x) = 0.5x2 5x + 3.5

g(2.8)

Example 5: Phil walks into his local gym and wants to sign up for a membership. The sales representative explains that a gym membership costs $30 per month, along with a one time registration fee of $99.

a) Write a function that represents the total amount of money f(m) that he has spent on his gym membership after m months.

b) How much money will he have spent after 6 months?

Example 6: Lauren bought a $50 pre paid phone card. For every minute that she talks, $0.40 is deducted from the card.

a) Write a function that represents the amount of money f(m) she has left after talking m minutes.

b) How much money is left on her card after she talks 30 minutes?

c) What is an appropriate domain for this problem? Why?

Linear Functions (book 2.2 p. 69)

Objective: I can identify linear relations and functions. I can write linear equations in standard form.

Linear Function:

LinearNon-linear

Standard Form of a Linear Equations:

Example: Write the following in standard form. Identify A, B, and C.

a) y = 3x 4b) 7y 4x = 9c)d)

In standard form it is very easy to find the x and y intercepts and graph the equation.

Ex. Find the x and y intercepts and graph3x 2y = 6

Sometimes the equation wont be in standard form so convert then find the x and y intercepts.

Ex. Find the x and y intercepts and graph.

a) 5y = 15x 45b)

Ex. The Ohio State Fair charges $8 for admission and $5 for parking. After Joey pays for admission and parking, he plans to spend all his remaining money at the ring game, which costs $3 per game.

a) Write an equation representing the situation

b) How much did Joey spend at the fair if he paid $6 for food and drinks and played the ring game 4 times?

Slope and Rate of Change (book 2.3 p. 76)

Objective: I can find the slope of a linear function given two points, a graph, table, and an application problem.

What do you remember about slope?

Slope/Rate of change:

Example 1: Find the slope given the following information.

a)b) (1, -4) and (-3, -8)c) 3x + 2y = 8

d)e) (2, -5) and (3, 1) f) 4x 5y + 2 =12

Example 2: Find the slope given the following information

a)b)

Special Cases

Vertical Lines:Horizontal Lines:

Example 3: Find the rate of change given the table below. Describe what the rate of change represents in this problem.

Time Driving (h)

Distance Traveled (mi)

2

80

4

160

6

240

Example 4: If you drive 40 miles in 1 hour, and on another trip you drive 85 miles in 2 hours. What is your average speed?

Example 5: Your dog enters a race against a rabbit. At t = 2 seconds of the race the distance d = 21.8 feet. At t = 14 seconds your dog is at a distance of d = 153 feet. What is his rate of change in feet per second?

Example 6: The table below shows the average yearly cost of gasoline over the last 6 years. What is the rate of change and what does it represent?

year

2009

2010

2011

2012

2013

2014

Average Price per Gallon

$2.73

$2.89

$3.05

$3.21

$3.37

$3.53

Writing equations of lines (book 2.4 p. 83)

Objective: I can write the equation of a line given a graph, the slope and a point, and two points.

Standard Form:

Slope-intercept form:

Point-slope form:

How to write the equation of a line

1.

2.

3.

Example 1: Write the equation of the line in slope-intercept form, given the following information. For example b, write answer in standard form too.

a) (0, -6) and (-4, 10)b) (-2,3) and (4, 5)

Example 2: Write the equation of the line in slope-intercept form, given the following information. For example b, write answer in standard form too.

a) slope = -4b) slope: 3

passes through ( 0, 2)passes through (-2, 4)

Example 3 (special cases): Write the following equations in slope intercept form.

a) slope = 0 and the line goes through (-3, 4)b) slope = undefined and the line goes through (1, -5)

Example 4: Write the equation of the line using the following graph

Writing Equations of Lines (book 2.4 p. 83)

Objective: I can write the equation of a line given a parallel or perpendicular line and from an application problem.

Ex. Write the equation of the line that is parallel to and passes through the point (-3, 1).

Ex. Write the equation of the line that passes through: (2, 11) perpendicular to the line passing thru: (1, 1) & (5, 7)

Ex. Each week, Carmen earns a base pay of $15 plus $0.17 for every pamphlet she delivers.

a) Write an equation that can be used to find out how much Carmen earns a week.

b) How much will she earn the week she delivers 300 pamphlets?

Ex. Whenever a sink overflows, you call the neighborhood plumber to snake the pipes. His fee varies linearly with the amount of time that he has to work. If he works for 20 minutes the fee is $24, and if he works an hour, his fee is $32.

a) Write an equation for the cost in term of hours.

b) What is the fee if Sam uses the plumber for 40 minutes?

Linear Regression (book 2.5 p. 92)

Objective: I can write the equation of the line of best fit given a set of data.

Line of Best Fit:

Example 1: The table shows the percent of US households with at least one personal computer.

Year

1984

1989

1993

1997

2001

2003

Percent

8.2

15.0

22.8

36.6

56.3

61.8

**Hint: whenever you are dealing with years, you need to change those values to smaller numbers. Make the first year 0, and then every other year can be re written as "years after 1984"

Year

0

5

9

13

17

19

Percent

8.2

15.0

22.8

36.6

56.3

61.8

a) Find the line of best fit

b) Use your equation to predict the percentage of

households that will have a personal computer.

Example 2: Fred is analyzing the sales of his company. He created the table below to show the sales over the last 6 years.

Year

2003

2004

2005

2006

2007

2008

2012

Sales ($ Millions)

31.2

34.6

18.9

37.7

41.3

45.1

?

a) Find a function S(t), where t is time in years since 2003, and S(t) represents sales in millions of dollars.

b) What is the slope of this function, and what does it represent?

c) Using this model, what do you predict the sales will be in 2012?

Example 3: This table shows the relationship between a class size and average grade in that class.

Class Size

16

19

24

26

27

29

32

35

Class Average

81.2

80.6

82.5

79.9

78.6

79.3

77.7

?

a) Write a function C(s) where s is class size, and C(s) represents the class average.

b) What is the slope of the function and what does it represent?

c) Using this model, what do you predict the class average will be when there are 35 students in class.

Graphing Lines (book 2.8 p.117)

Objective: I can graph a line given an equation by hand and using a graphing calculator.

X

Y

Example 1: Graph the line

y = -2x + 4

How to graph in the calculator:

1. Push the "y=" button and enter your equation (Hint: your equation must be in the form y = _____ in order to enter it in your calculator).

2. Push the "graph" button to see what the graph should look like.

3. Push the "2nd" button and then the "graph" button to access the table that goes with the graph.

Trouble shooting:

- to get a 10 x 10 grid push "zoom" and then 6 "standard".

- If you get an error message, make sure that the word "plot" is not highlighted in your "y=" window.

Example 2: Graph the lineExample 3: Graph the line

y = 3x 5 3y x = 6

Example 4: Graph the lineExample 5: Graph the line

4y + 3x = 8 y = -x + 4

Example 6: Joe has