Chapter  2:  Linear  Equations  and  Functions            

Chapter 2: Linear Equations and Functions Assignment Sheet

Date Topic Assignment Completed 2.1 Functions and their

Graphs and 2.2 Slope and Rate of Change 2.1 and 2.2 Homework Day 1

2.1 Functions and their Graphs and 2.2 Slope and Rate of Change 2.1 and 2.2 Homework Day 2

MATCH THE GRAPH SLOPE ACTIVITY

Complete the Match the Graph worksheet for homework.

2.3 Quick Graphs of Linear Equations and 2.4 Writing Equations of Lines Homework 2.3 and 2.4

More on 2.4 Why write equations of lines? pg. 97 #59-65

More on 2.4 Practice 2.4C (19-24) 2.5 Scatter Plots and

Correlation - Use Hoola-Hoop activity to collect linear data. NO HOMEWORK

Regression Lines using the Graphing Calculator 2.5 Homework

2.6 Linear Inequalities in Two Variables 2.6 Homework

2.7 Graphing Piecewise Functions 2.7 Homework (day 1)

2.7 Graphing and Writing Piecewise Functions 2.7 Homework (day 2)

Absolute Value Exploration with Desmos Complete the activity in class.

2.8 Absolute Value Graphs 2.8 Homework Review Multiple Choice Review Worksheet

Review pg. 130 # 1-25 pg. 133 # 1-24, 27

EXAM Chapter 2 Keystones Worksheet

6

4

2

-2

-4

-6

-5 5

2

-2

2

-2

2

-2

2

-2

s y( ) = 2

2.1- Functions and Their Graphs and 2.2- Slope and Rate of Change RECALL from Algebra I: coordinate plane – Cartesian plane or x-y plane ordered pairs – points in the form (x, y) x-coordinate – first value in an ordered pair,

• usually referred to as the __________________ • also called the ________________________

y-coordinate – the second value in an ordered pair,

• usually referred to as the ___________________ • also called the _______________________

Slope - a numerical value that represents the steepness of a line (m)

Slope = m = vertical change risehorizontal change run

=

Slopes of Graphs

The slope of a line passing through the points ),(),( 2211 yxandyx is

12

12

xxyy

m−−

=

Find the slope of the line passing through the points and tell whether it rises, falls, is horizontal, or vertical.

a.) (4, 2) (-18, 1) b.) (-7, 3) (-2, 3) c.) 1 3, 1 , 25 5

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

function – a set of ordered pairs for which there is exactly one y-value for each x-value. A function passes the _____________________________. Circle the graphs that represent a function. Applications: 1. You work at the local dairy queen and get paid $8.50 per hour. An equation modeling the money that you earn would be y = $8.50x where y equals the amount you get paid for working x hours. Create an x-y table of values and graph them below. x (hours) 1 2 4 6 8 10 y (dollars) Connect the points with a line. What does the slope of the line represent?

2. A water park slide drops 8 feet over a horizontal distance of 24 feet. a. Find its slope. b. Find the drop over a 54-foot section with the same slope. 3. The slope of a road, or grade, is usually expressed as a percent. For example, if a road has a grade of 3%, it rises 3 feet for every 100 feet of horizontal distance.

a. Find the grade of a road that rises 75 feet over a horizontal distance of 2000 feet.

b. Find the horizontal length of a road with a grade of 4% if the road rises 50 feet over its length.

4. The heights and ages of the players on a basketball team are shown in the graph below. Is height a function of age? 5. A cyclist maps her ride using an iPhone App that provides her with graphs of the elevation along the ride. She notices a pretty steep section around the 5-mile point.

a.) Use the graphs below to determine the grade of the road that she traveled near the 5-mile point.

b.) Use the graph above to determine the average speed of the cyclist during the first 20 minutes of her ride. SAT Type Problem: Find the value of k so that the line through the given points has the given slope.

1.) (5, k) and (k, 7), m=1 2.) (-2, k) and (k, 4), m=3

HOMEWORK 2.1/2.2 DAY 1 Read the article at http://www.nasdaq.com/article/could-t-mobile-really-pass-att-in-mobile-subscribers-cm391189 describing the rise of T-mobile. Under the heading “How big is the gap between the two?” there is a description of how quickly T-mobile is growing relative to Sprint and AT&T. Use the data in the paragraph to create a graph of the three companies starting with the end of the second quarter of this year (July, 2014) showing their total customers and projected rates of growth. Does the prediction of T-mobile passing AT&T in a little over two years seem plausible? Summarize the data from the article: Summarize the rates of growth listed in the article: Create a graph for all three carriers. Plot time on the x-axis and number of customers on the y-axis.

MATCH THE GRAPH SLOPE ACTIVITY An exploration with the CBR units and graphing calculators

Calculator Instructions :

• APPS • Easy Data • Setup • 1 (Dist), enter • change units to feet, OK • Setup • 3 (Distance Match) • Start • OK • Next • Start (when you are ready to match the graph)

1.) Sketch a graph of the graph that you were trying to match a.)What variable is being represented on the y-axis ? (label this) ____________ units ?_________ b.)What variable is being represented on the x-axis ? (label this) ____________ units ?___________ c.) What does the y-intercept represent ? _____________________________________ Try a second time to match the graph that you sketched by hitting the “retry” key on the calculator.

2.) Hit the “new” button on the graphing calculator to generate a different graph and then sketch the graph below Hit the “retry” button to perfect your matching of the graph. Discuss the different properties of your graph below : 3.) Hit the “new” button on the graphing calculator to generate a different graph and then sketch the graph below Hit the “retry” button to perfect your matching of the graph. Discuss the different properties of your graph below:

FOLLOW UP QUESTIONS: 1.) What does the slope of the line represent in each graph?

a.) What does a positive slope represent in the context of your graph? _______________________________________________________

b.) What does a negative slope represent in the context of your graph?

c.) What does a zero slope (horizontal line) represent in the context of your graph? _______________________________________________________

d.) If the slope of one line is steeper than another, what does that indicate in the context of your graph?

___________________________________________________________ 2.) How did you know how far from the wall to stand when you originally looked at the graph? ____________________________________________________________

2.1/2.2 Homework Day 2 Identify the domain and range. Graph the relation. Then tell whether the relation is a function.

Function? Domain: _____________________ _____________ Range: ______________________ Use the vertical line test to determine whether the relation is a function. Function? ____________ Evaluate the following functions.

�

f (x) = −23x 2 − x + 5; f (6) f (x) = −3 + 4x; f −

12

⎛ ⎝

⎞ ⎠

Estimate the slope of the line. m = _____________ m = ___________ m = _________

Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal or vertical.

(-10, -12) and (2, -6) (-1,4) and (-2,4) (0,

�

72

) and ( 2,

�

52

)

Which of the above lines is the steepest? _______________________________________ Application You are in charge of building a wheel chair ramp for a doctor’s office. Federal regulations require that the ramp must extend 12 inches for every 1 inch of rise. The ramp needs to be a height of 18 inches.

a.) How  far  should  the  end  of  the  ramp  be  from  the  base  of  the  building?  

b.) Use  the  Pythagorean  Theorem  to  determine  the  length    of  the  ramp.   Find the value of k so that the line through the given points has the given slope.

(-3, 2k) and (k,6), m=4 (9, -k) and (3k, -1), m =

�

−13

2.3- Quick Graphs of Linear Equations and 2.4- Writing Equations of Lines Notes The SLOPE INTERCEPT form of a line is: STANDARD FORM OF A LINE: Ax + By = C where A, B, and C are ____________________ and A ___________ The X-INTERCEPT of a line is where _______________________________________. To find the x-intercept, __________________________________________________.

�

y = −2x + 3

�

y =23x − 3

�

y = −25x +

12

x-intercept: __________________ x-intercept: __________________ x-intercept: __________________ Standard Form:_______________ Standard Form:_______________ Standard Form:_______________ What formula can you use to generalize the slope of a line if the equation is given in standard form? m = ______ 5x + 3y = -15 slope:______ y-intercept:___________ x-intercept:_______

�

y = 3x

�

y = 7

�

x = −4

Graph then write an equation of the line that passes through the point ( -3, 4) and

32=m

A.) Slope-Intercept Form: y = mx + b

B.) Point-Slope Form: 1 1( )y y m x x− = −

C.) Standard Form: Ax + By = C

1) Write  the  equation  of  a  line  that  passes  through  the  point  (2,3)  and  a  slope  of   −12    

2) Write the equation of the ling that passes through the point (7,-4) and m = 25

Given Two Points: Write the equation of the line that passes through )4,3()1,2( and−−

1) Write the equation of the line shown. State your answer in slope-intercept Form.

2) Write  an  equation  of  the  line  that  passes  through  the  points  (2,0)  and  (4,-­‐6)   3) Given  the  points  (-­‐8,8)  and  (0,1),  write  the  equation  of  the  line.   Writing Equations of Parallel and Perpendicular Lines Parallel Lines: Two lines are parallel if and only if they have the same ________________________. Perpendicular lines: Two lines are perpendicular if and only if their slopes are ___________________________ AND _________________________ of each other. Key Questions: All types of vertical lines are parallel to what types of lines? _______________________ All types of vertical lines are perpendicular to what types of lines? _________________

Graph and use the slope formula tell whether the lines are parallel, perpendicular or neither. Line 1: through )1,3()3,3( −− and

Line 2: through )3,2()3,2( and−−

Write an equation of the line that passes through (2, -3) and is parallel to the line that passes through (3, 5) and (-1, -3)

Write an equation of the line that passes through (3, 2) and is perpendicular to the line 23 +−= xy

60

55

50

45

40

35

30

25

20

15

10

5

-5

5 10 15 20

Write an equation of a line that passes through (3, -5) that is a) perpendicular and b) parallel to the line x = 4. Then graph each line on the graph provided.

a.) perpendicular  

b.) parallel

Why write equations of lines? Create a graph to represent the data given below regarding college tuition for out of state students at the University of Michigan. Based on the observed rate of growth, what will it cost to attend the University of Michigan ten years from now? Cost for two semesters year $12,000 1994 $28,000 2006

Thousands Of $

years since 1990

Direct Variation The variables x and y show direction variation given the following equation is true: __________________ where __________________. EX: y = 5x k is the __________________ of _____________________.

• Y-intercept is equal to zero • The graph goes through the origin

EX: The variables x and y vary directly, and when y = 12, x = 4. Write then graph the equation that relates x and y. Find y when x = 5 EX: Using what you k now about the equations of lines and the graphs of direct variation, decide if the equation represents direct variation. A.) 2y – 5x = 0 B.) 3y – 7 = 10x EX: Write and graph a direct variation equation that has the given ordered pair as a solution: (6, -2)

EX: Tell whether the data shows direct variation.

14 Karat Gold Chains (1 gram per inch) Length, x (inches)

16 18 20 24 30

Price, y (dollars)

288 324 360 432 540

EX:  In  a  recipe  for  cookies,  the  amount  of  flour  varies  directly  wit  the  amount  of  butter.    The  recipe  calls  for  4  cups  of  flour  and  2/3  cup  of  butter.    What  is  the  constant  of  variation?    Write  and  equation  representing  how  the  flour,  f,  varies  directly  wit  the  butter,  b.   EX:  The  amount  of  sales  tax  that  you  pay  in  the  state  of  Illinois  varies  directly  wit  the  price  of  the  item.    The  sales  tax  on  clothing  is  6.25%.    Write  an  equation  involving  direct  variation  to  represent  the  tax  charged  for  an  item  of  clothing.    What  does  k  represent  if  you  equation?

2.3 and 2.4 Homework  

1.) Graph  the  following  equations,  find  their  x-­‐intercept,  and  then  write  the  equation  in  standard  form.  

�

y = 5x +1

�

y = −54x + 2

�

y =34x −

14

x-intercept: __________________ x-intercept: __________________ x-intercept: __________________ Standard Form:_______________ Standard Form:_______________ Standard Form:_______________ 2.) Find  the  slope  of  the  equation  5x  –  2y  =  20.    Find  the  intercepts  and  then  graph    

 the  equation.     Slope: ________________ x-intercept: _____________ y-intercept: _____________

       

3.) Write  the  equation  of  the  line  that  passes  through  the  point  (7,-­‐4)  and  has  a  slope  of  

�

25.  

4.)  Write  the  equation  of  the  line  that  passes  through  the  point  (-­‐6,5)  and  has  a  slope  of  0.   5.) Write  an  equation  of  the  line  that  passes  through  (1,  -­‐1)  and  is  perpendicular  to  the  line  

�

y = −12x + 6 .  

6.) Write  an  equation  of  the  line  that  passes  through  (6,  -­‐10)  and  is  perpendicular  to  the  line  that  

passes  through  (4,-­‐6)  and  (3,-­‐4).     7.) Write  an  equation  of  the  line  that  passes  through  (2,-­‐7)  and  is  parallel  to  the  line  x=5.  

8.) Write  an  equation  of  the  line  that  passes  through  (4,6)  and  is  parallel  to  the  line  that  passes  through  (6,-­‐6)  and  (10,-­‐4).    

9.) Write an equation of the line shown. 10.) Write the equation of the line that passes through the points (2,0) and (4,-6)

 

HOOLA HOOP CHALLENGE In class, we collected data for the amount of time for a ‘hoola hoop pass.’ Please list the data in the given table. # of People

Time Elapsed

1.) Which variable is independent? 2.) Which variable is dependent? 3.) Which axis should the independent variable be on? 4.) Which axis should the dependent variable be on?

PLOT the above ordered pairs using the scatter plot feature of your graphing calculator. Describe the correlation of the data:___________________________________ Derive the equation of the best fit line from your graphing calculator:____________________ FOLLOW UP QUESTIONS:

1.) Interpret the slope of your regression line in terms of # of people and time elapsed.

2.) Interpret the y-intercept of your regression line in terms of # of people and time elapsed.

3.) From the information collected, can you predict how long it would take 10 people to complete the hoola hoop pass? Is this an example of interpolation or extrapolation?

4.) From the information collected, can you predict how long it would take 30 people to complete the hoola hoop pass? Is this an example of interpolation or extrapolation?

5.) How many people could have been involved in the activity if the hoola hoop pass took 72 seconds?

6.) Do you find any real life conditions in this experiment that will impact our predictions?

2.5- Correlation and Best-Fitting Lines Scatter Plot: a graph of a set of data pairs (x,y) Positive Correlation: y tends to _____________________ as x increases Negative Correlation: y tends to _____________________ as x increases NO Correlation: the points show ________________________________

1.) Think of two variables that have a positive correlation: 2.) Think of two variables that have a negative correlation:

REGENTS SCORE VS. STUDY HOURS

Calculate the Best Fit Line by Hand: y = ___________________ Find the BEST best-fitting line using your graphing calculator : y = _______________ The independent variable is ___________ and the dependent variable is ___________ Use interpolation to predict a student’s regents score if they studied for 4 hours _______ Interpret the slope of the regression lines in terms of hours of study and regents score.

Study Hours Regents Score

3 80 5 90 2 75 6 80 7 90 1 50 2 65 7 85 1 40

7 100

Creating a Scatter Plot and Regression Line in your Calculator ENTER THE DATA: STAT 1:EDIT ENTER Enter the data into your lists SET UP THE SCATTER PLOT: 2ND Y= ENTER Set up Plot 1 as shown below: SET UP THE WINDOW: 2ND WINDOW Choose window settings that fit the min and max values of your data in the lists. Remember the x-values are in List 1 and the y-values are in List 2. CREATE THE SCATTER PLOT: GRAPH CREATE THE LINEAR REGRESSION MODEL: STAT CALC 4:LinReg(ax+b) ENTER ENTER Record your equation, then go to the Y= screen and graph the equation. Y= enter the equation (the variable “x” is just below the MODE button) GRAPH

2.5 Homework The table below gives the average life expectancy (in years) of a person based on various years of birth. Let x represent the number of years since 1900. (x = 10 represents the year 1910)

Year of Birth (x) 0 10 20 30 40 50 60 70 80 90 Life Expectancy (y) 47.3 50 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4

a.) Graph  the  data  below  (be  sure  to  scale  and  label  your  axes  appropriately).  Draw  in  an  estimate  of  the  best  fit  

line.    Find  two  points  on  the  line  and  estimate  the  equation  for  the  best  fit  line  without  a  calculator.   Equation of best fit line from graph (no calculator): y = _____________________________ SHOW WORK! b.) Graph  a  scatterplot  of  the  data  on  your  graphing  calculator.  Find  the  equation  of  the  best  fit  line  from  your  

graphing  calculator.    Equation of best fit line from calculator: y = __________________________________

c.) Use  the  equation  from  your  calculator  to  predict  the  life  expectancy  for  someone    born  in  2010.  

 d.) The  independent  variable  is  ______________.    The  dependent  variable  is______________   e.) Interpret  the  slope  in  terms  of  year  of  birth  and  life  expectancy.  

 

The Stroop Test One of the main uses of data is to make predictions about real-world situations. We are going to perform an experiment from cognitive psychology, which is the branch of psychology that tries to understand and explain how the human brain works. The experiment is named after the man who first performed it, J.E. Stroop.

• Each student will look at a list of words written in color – red, green, black, or blue. Each list is different in length.

• The student will be asked to say the color of the ink for each word. Three timers

will record the time needed to complete each list and the average noted.

• Two different lists will be used. One in which the color of the ink matches the color of the word, for example, red written in red ink; and a second where the color of the ink does not match the color of the word, for examples, red written in blue ink. The first type is called matching, the second, non-matching.

Answer the following questions before we begin:

1.) What do you think we’ll find when we perform these experiments? How will the matching data differ from the non-matching data ?

2.) What question would a cognitive psychologist be trying to answer by performing these experiments ?

Matching Non-Matching

List Length Time

List Length Time

Graphing Calculators Use the graphing calculator to perform a linear regression of time vs. list length for matching data. Repeat for the non-matching data.

Matching: _________________ Non-matching: ______________________

Discussion

1.) Interpret the slope and y-intercept of the matching linear regression line. In what units are they measured? What would these points mean to a cognitive psychologist?

2.) Making predictions: Use your equation to estimate how long it would take to name

the colors in a list of 10 matching words. How long would it take for 25 words? Which is an example of extrapolation and which is an example of interpolation ?

3.) Explain why you must be careful when extrapolating.

4.) Interpret the slope and y-intercept of the non-matching linear regression line. In

what units are they measured? What would these points mean to a cognitive psychologist?

5.) What conclusions would a cognitive psychologist draw from this experiment?

2.6-­‐  Linear  Inequalities  in  Two  Variables   Warm Up: Check whether the given ordered pair is a solution of 532 −≥− yx 1.) (-2, -5) 2.) (2,1) 3.) (-4, -1) ACTIVITY:

1.) Solution (color): Not a Solution (color):

2.) Test the following ordered pairs in the inequality 1≥+ yx

Then plot them in accordance with the color-coding that you defined in 1. (0, 0) ( 2,0) (4,0) (-2,0) (-4, 0) (0,2) (2,2) (4,2) (-2,2) (-4, 2) (0, 4) (2,4) (4,4) (-2, 4) (-4,4) (0, -2) (2,-2) (4, -2) (-2,-2) (-4, -2) (0, -4) (2, -4) (4,-4) (-2, -4) (-4,-4)

What is the case about the points on the line x + y = 1? Describe a general strategy for graphing an inequality in two variables. Graph the following inequalities: 425 −≤− yx y < 2x y < -2 1≤x

2.6  Homework   Check whether the given ordered pairs are solutions of the inequality.

�

y < −9x + 7; (3,−8)

�

x ≤ −5; (−5,1) Match the inequality with its graph.

�

1.) 2x − y ≥ 4 2.) − 2x − y < 4 3.) 2x + y ≤ 4

Graph the following inequalities on the coordinate plane

�

9x − 2y ≤ −18 6x ≥ −13y x + y < 0 5x > −20

Write the inequality whose graph is shown.

2.7-­‐  Piecewise  Functions    What  is  a  piecewise  function?            1)  

f −1( ) =

f 5( ) =

f −4( ) =

2)  

  f 2( ) =

f −1( ) =

f 0( ) =

3)    

f 6( ) =

f 2( ) =

f −10( ) =

4)  

f −4( ) =

f 0( ) =

f 3( ) =

5)  

f −3( ) =

f 4( ) =

2.7  Homework  Day  1   Evaluate the function for the given value of x. f(-4)= f(-2)= f(0)= f(5)= Graph the following functions.

2.7  Homework  Day  2    

Write equations for the piecewise functions whose graph is shown.

Exploring Graphs of Absolute Value Functions in Desmos Start up: 1) Open the Desmos app on your iPad 2) Enter in the equation y = a x − h + k 3) Add sliders to all three variables a, h, and k Begin: Adjust the sliders to that a = 1, h = 0 and k = 0. The sliders a, h and k represent the constants in the equation q(x). 1) Write the equation of the graph. ____________________________ (This is the parent function for the family of absolute value functions) 2) Fill out the table of values that corresponds with the function by reading the values from the graph. x -2 -1 0 1 2 f(x) 3) What is the shape of the graph? 4) Where is the vertex of the graph? Name the coordinates. 5) Adjust the sliders so that a = 1, h = 2, and k = 3. Where is the vertex of the new graph? _________________________________________________________________ Write the new equation of the graph: __________________________________________________________________ What are the slopes of the branches of the new graph? ____________________________________________________ Describe the translation of the parent function to attain the new graph________________________________________ 6) Adjust the sliders so that a = 4, h = 2, and k = 3. Where is the vertex of the new graph? ________________________________________________________________ Write the new equation of the graph: _________________________________________________________________ What are the slopes of the branches of the new graph? ___________________________________________________ Describe the translation of the parent function to attain the new graph________________________________________ 7) Adjust the sliders so that a = -4, h = 2, and k = 3. Where is the vertex of the new graph? _________________________________________________________________ Write the new equation of the graph: __________________________________________________________________ What are the slopes of the branches of the new graph? _____________________________________________________ Describe the translation of the parent function to attain the new graph_________________________________________

8) Adjust the sliders so that a = 3, h = -4, and k = -1. Where is the vertex of the new graph? _________________________________________________________________ Write the new equation of the graph: __________________________________________________________________ What are the slopes of the branches of the new graph? ____________________________________________________ Describe the translation of the parent function to attain the new graph_________________________________________

9) Adjust the sliders so that a =

�

12

, h = -2, and k = -5.

Where is the vertex of the new graph? _________________________________________________________________ Write the new equation of the graph: __________________________________________________________________ What are the slopes of the branches of the new graph? _____________________________________________________ Describe the translation of the parent function to attain the new graph_________________________________________

10) Adjust the sliders so that a =

�

−15

, h = -3, and k = 4.

Where is the vertex of the new graph? _________________________________________________________________ Write the new equation of the graph: __________________________________________________________________ What are the slopes of the branches of the new graph? _____________________________________________________ Describe the translation of the parent function to attain the new graph_________________________________________ Select the blue dot on the slider that corresponds with parameter a. Describe the two things that happen to the graph as you change parameter a. 1.)_______________________________________________________________________________ 2.) ______________________________________________________________________________ What special property does the graph have when a = 0? ____________________________________ Select the blue dot on the slider that corresponds with parameter h. What happens to the graph as you change the parameter h? _________________________________ ________________________________________________________________________________ What special property does the graph have when h = 0? ____________________________________ ________________________________________________________________________________

Select the blue dot on the slider that corresponds with parameter k. What happens to the graph as you change the parameter k? _________________________________ ________________________________________________________________________________ What special property does the graph have when k = 0? _____________________________________ ________________________________________________________________________________ Describe your findings. What does each variable a, h, and k represent? How does changing each change the look of the graph? Be specific and use examples.

2.8 Absolute Value Functions (Group Work Discovery Activity) General Graphing Form: y = a x − h + k Use the table of values below in order to graph .xy = X -3 -2 -1 0 1 2 3 y

Use the table of values to graph the following equations 1.) 1+= xy X -3 -2 -1 0 1 2 3 y

A.) Describe the translation of this graph from y = x :

It is a _____________ shift of _______. B.) Left slope: ________ Right slope: _______ C.) Vertex: __________ D.) Axis of Symmetry: ______

2.) 1+= xy X -4 -3 -2 -1 0 1 2 y A.) Describe the translation of this graph from y = x It is a _____________ shift of _______. B.) Left slope: ________ Right slope: _______ C.) Vertex: __________ D.) Axis of Symmetry: ______ 3.) 1−= xy X -3 -2 -1 0 1 2 3 y A.) Describe the translation of this graph from y = x It is a _____________ shift of _______ B.) Left slope: ________ Right slope: _______ C.) Vertex: __________ D.) Axis of Symmetry: ______

4) 1−= xy x -2 -1 0 1 2 3 4 y A.) Describe the translation of this graph from y = x : It is a _____________ shift of _______ B.) Left slope: ________ Right slope: _______ C.) Vertex: _________ D.) Axis of Symmetry: ______ 5) xy −= x -3 -2 -1 0 1 2 3 y

A.) Describe the translation of

this graph from y = x : It is a _____________ about the _______ B.) Left slope: ________ Right slope: _______ C.) Vertex: __________ D.) Axis of Symmetry: ______

6.) xy 3= x -3 -2 -1 0 1 2 3 y

A.) How does this graph compare to y = x ?

B.) Left slope: ________ Right slope: _______ C.) Vertex: _________ D.) Axis of Symmetry: ______ 7.) xy

21=

X -6 -4 -2 0 2 4 6 y

A.) How does this graph compare to y = x ?

B.) Left slope: ________ Right slope: _______ C.) Vertex: _________ D.) Axis of Symmetry: ______

Using the equations from numbers 1-7 organize the equations into the appropriate categories. Original absolute value function: __________________ Horizontal Shift Graphs Vertical Shift Graphs 1. 1. 2. 2. Flipped Graphs Wider or More Narrow Graphs 1. 1. 2. In all of the above equations where are the slopes located? Predict the transformation for the following graphs then use a table to verify the graph by graphing with a table. 1.) 32 ++−= xy A.) Horizontal shift?

B.) Vertical Shift? C.) Direction opens? Wider or more narrow? X -5 -4 -3 -2 -1 0 1 y

Predict the transformation for the following graphs then use a table to verify the graph by graphing with a table. 2) 112 +−= xy A.) Horizontal shift? B.) Vertical Shift? C.) Direction opens? Wider or more narrow X -2 -1 0 1 2 3 4 y Predict the transformation for the following graphs then use a table to verify the graph by graphing with a table. 3) 2

21 −= xy A.) Horizontal shift?

B.) Vertical Shift? C.) Direction opens? Wider or more narrow? X -3 -2 -1 0 1 2 3 y

2.8  Homework   Graph the function and describe the translation of the parent function

�

y = x to attain the graph.

�

y = 6 x − 7 y = x + 9 y = − x − 8 +1 Translation of

�

y = x Translation of

�

y = x Translation of

�

y = x ______________________ _______________________ _____________________

�

y = − x + 2 +11 y =13x − 3 + 4 y =

12x + 6

Translation of

�

y = x Translation of

�

y = x Translation of

�

y = x

____________________________ _________________________ _____________________

Write an equation of the graph shown.

1 Algebra II Chapter 2 Review Sheet 1. Given g(x) = x2 + 3x − 5 , what is g(−2) ?

a. -7 b. -1 c. 5 d. -3 e. -15

2. What is the slope of the line passing through the points (-5, 4) and (-1, 8)? a. -2/3 b. -1 c. 3/2 d. 1 e. 2/3 3. What is the slope of the line passing through the points (2, 5) and (-4, 3) ? a. 1/3 b. 3 c. -3 d. -1/3 e. 1 4. Which of the following lines is the steepest? a. Line 1 through (4, 5) and (3, 2) b. Line 2 through (6, 4) and (3, 2) c. Line 3 through (-2, 7) and (6, 6) d. Line 4 through (-1, 3) and (4, 5) e. Line 5 through (-2, -4) and (1, 2) 5. Thie line that passes through the points (3, 0) and (-5, 8) : a. is vertical b. falls c. is horizontal d. rises e. doesn’t exist 6. A ladder 40 feet in length that hits the wall at a height of 24 feet has a slope of: a. ¾ b. 4/3 c. 3/5 d. 5/3 e. 4/5

7. What is the slope of the line y = − 15x − 7 ?

a. 1/5 b. -7 c. -5 d. -1/5 e. 7

8. What is the y-intercept of the line y = 23x + 4

5 ?

a. 2/3 b. 4/5 c. -4/5 d. -2/3 e. 5/4

4

2

–2

–4

–6

5

9. What is the x-intercept of the line y = − 14x −16 ?

a. -1/4 b. -16 c. 64 d. 4 e. -64 10. What is the y-intercept and the x-intercept of the line y = −5x −15 ? a. y-intercept = 15, x-intercept = -5 b. y-intercept = -15, x-intercept = -3 c. y-intercept = 15, x-intercept = -3 d. y-intercept = 15, x-intercept = 3 e. y-intercept = -5, x-intercept = -15 11. Which function is represented by the graph shown at right? a. −2x − 4y = 8 b. 2x + 4y = 8 c. 2x − 4y = −8 d. −2x − 4y = −8 e. 2x − 4y = 8 12. What is the equation of the line that passes through the point (1,5) and has a slope of -3? a. y = 3x + 8 b. y = 3x − 2 c. y = −3x − 8 d. y = −3x + 8 e. y = −3x + 2 13. What is the equation of the line that passes through the point (-2, -3) and has a slope of 4? a. y = 4x + 5 b. y = 4x +11 c. y = −4x + 5 d. y = 4x −11 e. y = 4x − 5 14. What is the equation of the line that passes through the point (-1, 7) and is parallel to the line

y = −2x −1?

a. y = 12x + 15

2 b. y = 1

2x + 13

2 c. y = 2x + 5

d. y = 2x − 9 e. y = −2x + 5

15. What is the equation of the line that passes through the point (6, -3) and is perpendicular to the

line y = 13x + 4?

a. y = 3x +15 b. y = 3x −15 c. y = −3x +15

d. y = 13x − 5 e. y = 1

3x −1

16. If y tends to increase as x increases on a scatter plot, what is the correlation of the paired data? a. positive b. negative c. relatively no correlation d. undefined e. none 17. Which is the best fitting line for the data shown?

X 1 2 3 4 5 Y 4.2 3.8 3.5 2.7 2.2

a. y = 0.51x + 4.81 b. y = −0.051x + 4.8 c. y = 5.1x + .481 d. y = −0.51x + 4.81 e. y = −0.0051x + 4.81 18. Which of the ordered pairs is a solution of the inequality 2x − 5y ≤ 8? a. (2, -5) b. (0, -3) c. (-1, -2) d. (5, 0) e. (-2, -5) 19. Which of the ordered pairs is not a solution of x < −y + 3? a. (-3, -5) b. (0, 2) c. (-2, 0) d. (2, 2) e. (-1, 3)

20. Given f (x) =4x +1, x < −12x − 3, x ≥ −1

⎧⎨⎪

⎩⎪, what is f (−1)?

a. -5 b. 3 c. -1 d. -3 e.

5

21. Given f (x) =−5, −3< x < 03, 0 ≤ x ≤17, 1< x < 4

⎧

⎨⎪

⎩⎪

, what is f (1)?

a. -5 b. 1 c. 0 d. 7 e. 3

8

6

4

2

–5 5

4

2

–2

–4

–5 5

22. Which function is represented by the graph shown at right? a. y = 2x −1 b. y = 2x +1 c. y = −2x −1 d. y = −2x −1 e. y = 2x −1 23. Which statement is true about the graph of the function y = x − 7 + 3 ? a. Its vertex is at (0, 10) b. Its vertex is at (-7, 3) c. Its vertex is at (3, 7) d. Its vertex is at (10, 0) e. Its vertex is at (7, 3)

24. Given f (x) =−x, −1≤ x ≤ 0x, 0 < x ≤1

−2x, 1< x ≤ 2

⎧

⎨⎪

⎩⎪

, what is f 12

⎛⎝⎜

⎞⎠⎟ ?

a. -1 b. 2 c. ½ d. -2 e. -1/2

25. Which function is represented by the graph shown? a. y = 3x + 2 b. y = 2 − 3x c. y = − 3x + 2 d. y = −3x + 2 e. y = 3x + 2 26. What is the equation of the line passing through the point (-1, 3) with a slope of 1/2?

a. (y − 3) = 12(x +1) b. (y + 3) = 1

2(x +1)

c. (y − 3) = 12(x −1) d. (y + 3) = 1

2(x −1) e. none of the above

27. Determine the Standard Form of the equation y = −2x + 3 . a. (y − 3) = −2(x + 0) b. (y + 3) = −2(x + 0) c. 2x + y = 3

d. −2x + y = 3 e. y = 12x + 3

28. Which equation form does NOT describe a linear function? a. y = mx + b b. Ax + By = C c. y = x − h + k d. (y − y1) = m(x − x1)