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 / y2 / -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 won’t 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 / 2014Average 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 $32.
a) Write an equation for the cost in term of hours.
b) What is the fee if Sam uses the plumber for 40 minutes?
Linear Regression (book 2.5 p. 92)
Objective: I can write the equation of the line of best fit given a set of data.
Line of Best Fit:
Example 1: The table shows the percent of US households with at least one personal computer.
Year / 1984 / 1989 / 1993 / 1997 / 2001 / 2003Percent / 8.2 / 15.0 / 22.8 / 36.6 / 56.3 / 61.8
**Hint: whenever you are dealing with years, you need to change those values to smaller numbers. Make the first year 0, and then every other year can be re written as "years after 1984"
Year / 0 / 5 / 9 / 13 / 17 / 19Percent / 8.2 / 15.0 / 22.8 / 36.6 / 56.3 / 61.8
a) Find the line of best fit
b) Use your equation to predict the percentage of
households that will have a personal computer.
Example 2: Fred is analyzing the sales of his company. He created the table below to show the sales over the last 6 years.
Year / 2003 / 2004 / 2005 / 2006 / 2007 / 2008 / 2012Sales ($ 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 / 35Class 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 / YExample 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 a Starbucks gift card worth $100. The first day that Joe
used his gift card he bought a tall vanilla latte for $2.30. He liked that latte so
much that every time after that visit he bought a venti vanilla latte for $5.30 each.
a) Write a function that describes the situation above.
b) Graph the function.
c) What is the slope of your function and what does it represent?
Graphing Linear Inequalities (book 2.8 p.117)
Objective: I can graph linear inequalities using technology, and apply them in modeling problems.
What is an inequality?
**Important**
When you multiply or divide both sides of an inequality by a negative number you must FLIP the inequality sign.
Graphing Inequalities
Just like graphing lines, except you have to add a few things.
Example 1: Graph the linear inequalityExample 2: Graph the linear inequality
y 1.5x + 23x + 2y ≥ 12
Example 3: The Airbus A380 can seat up to 853 passengers. Suppose there are currently 632 passengers on board the airplane. Write and solve an inequality that indicates how many additional passengers are able to board.
Example 4: A farmer is going to plant tomatoes and peppers on his farm. Each acre of tomatoes costs $300 to plant, and each acre of peppers costs $100 to plant. His cost must be no more than $5,000.
a) Let x be the number of acres of tomatoes, and y be the number of acres of peppers. Write an inequality to represent the situation.
b) Graph the inequality.
c) What is the domain for the tomatoes?
d) What is the range for the peppers?
Piecewise Functions (book 2.6 p. 101)
Objective: I can evaluate and graph piecewise functions.
What is a piecewise function?
Example 1: Evaluate each function for the specified value.
a) f(0)b) f(-3)c) f(3)
Example 2:
a) f(0)b) f(-1)c) f(4)
Example 3: Graph the following piecewise function
Example 4:Example 5:
Solve systems by graphing, substitution, and elimination (book 3.1 p. 135 & 3.2 p. 143)
Objective: I can solve a system of linear equations by graphing, substitution, and elimination.
System of Equations:
Solution:
Example 1: What is the solution to the system?Example 2: Solve by graphing
Example 3:
3x + y = 11
x – 2y = 6
Substitution:
Example 1: Solve by substitution Example 2: Solve by substitution
5x – 3y = 233x + y = 5
y = -2x + 72x - y =10
Example 3: Solve by substitution
y = 2x – 1
4x + 2y = 15
Elimination:
Example 4: Solve by eliminationExample 5: Solve by elimination
2x + 4y = -193x =-5y + 6
-3x + 4y = -2-4x + 2y = 5
Example 6:
5x + 6y = -45
-10x -12y = 90
Application problems (book 3.1 p. 135 & 3.2 p. 143)
Objective: I can solve application problems using systems of equations.
Helpful tips:
Example 1: Lauren's bank account has a balance of $60. Every week she adds $10 to her account. Bailey's bank account has a balance of $120. Each week she withdraws $20.
a) Define your variables and write two equations that represent Lauren and Bailey’s accounts.
b) Graph each function in the coordinate plane.
c) When will Lauren and Bailey have the same amount of money in their accounts. This is called the break-even point.
d) What is the domain that represents Bailey having more money than Lauren.
Example 2. Libby borrowed $450 to start a lawn-mowing business. She charges $35 per lawn and incurs $8 in operating costs per lawn. How many lawns must she mow to make a profit?
Example 3: A movie theater sold a total of 200 tickets for a show- some adult tickets for $6 each and some students tickets for $4 each. If the theater took in $1040, how many of each type of ticket did they sell?
Example 4. At a park, there are 38 people playing tennis. Some are playing doubles, and some are playing singles. There are 13 matches in progress. A doubles match requires 4 players, and a singles match requires 2 players.
a) Write a system of equations to represent the number of matches going on.
b) How many matches of each kind is in progress?
Example 5: Angles A and B are supplementary and the measure of angle A is 18 degrees greater than the measure of angle B. Find the angle measures using a system of equations.
Solve systems using matrices (book 3.1 p. 135 & 3.2 p. 143)
Objective: I can solve systems with matrices using technology.
Matrix:
Element:
Dimensions of a Matrix:
Matrices can be used to solve systems by putting the systems in reduced row echelon form, which the calculator can do for you.
Reduced Row Echelon Form
A matrix is said to be in reduced row echelon form if
1.
2.
Ex:
How to enter into your calculator
3. Hit 2nd quit (mode)
To get back to the
Main screen.
1. Hit ______and 2. Enter the ______
Scroll over to edit of your matrix ( ______)
Use zero if a variable is missing.
4. Go back to the matices -5. Hit enter and there is x, y, and z!!!
Hit ______
Go to math and
Down to ______
Example: Solve each system using matrices.
a) b) c)
Matrices can make solving the application problems easier also!!! You just have to be able to set up the system in standard form.
Example: A store sold a total of 125 car stereo systems and speakers in one week. The stereo systems sold for a $112 each and the speakers sold for $18 each. The sales from these two items totaled $4224. How many of each type were sold? If c = number of stereo systems and t = number of speakers
Example: In the struggling economy, the demographics of two neighboring cities (Metropolis and Gotham) change. There are two main industry jobs in both cities: Manufacturing and Service. Metropolis increased its job market by creating 2 new service industries and was able to keep all 10 of its manufacturing industries. Meanwhile Gotham lost 2 service industries but did hold onto all 6 of its manufacturing industries. The total revenue in service and manufacturing in Metropolis was 560 million dollars last year and the total revenue in Gotham was 240 million dollars. How much is each industry worth in millions of dollars?
Example: Jackie and Cheryl work at a pizza shop. Cheryl earns $8.50 per hour and Jackie earns $7.50 per hour. During a typical week, Cheryl and Jackie earn 299.50 together. On this week, Jackie doubles her work hours, and the girls earn $412. How many hours does each girl work?