Chapter 2: Linear Equations and Functions Chapter 2: Linear Equations and Functions
Assignment Sheet
Date
Topic
2.1 Functions and their
Graphs and 2.2 Slope and
Rate of Change
2.1 Functions and their
Graphs and 2.2 Slope and
Rate of Change
MATCH THE GRAPH
SLOPE ACTIVITY
2.3 Quick Graphs of Linear
Equations and 2.4 Writing
Equations of Lines
More on 2.4 Why write
equations of lines?
More on 2.4
2.5 Scatter Plots and
Correlation - Use HoolaHoop activity to collect
linear data.
Regression Lines using the
Graphing Calculator
2.6 Linear Inequalities in
Two Variables
2.7 Graphing Piecewise
Functions
2.7 Graphing and Writing
Piecewise Functions
Absolute Value Exploration
with Desmos
2.8 Absolute Value Graphs
Review
Review
EXAM
Assignment
2.1 and 2.2 Homework Day 1
2.1 and 2.2 Homework Day 2
Complete the Match the Graph
worksheet for homework.
Homework 2.3 and 2.4
pg. 97 #59-65
Practice 2.4C (19-24)
NO HOMEWORK
2.5 Homework
2.6 Homework
2.7 Homework (day 1)
2.7 Homework (day 2)
Complete the activity in class.
2.8 Homework
Multiple Choice Review Worksheet
pg. 130 # 1-25
pg. 133 # 1-24, 27
Chapter 2 Keystones Worksheet
Completed
2.1- Functions and Their Graphs and 2.2- Slope and Rate of Change
6
RECALL from Algebra I:
4
coordinate plane – Cartesian plane or x-y plane
2
ordered pairs – points in the form (x, y)
-5
5
-2
x-coordinate – first value in an ordered pair,
• usually referred to as the __________________
• also called the ________________________
-4
-6
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
rise
=
horizontal change run
Slopes of Graphs
s ( y)
2
-2
2
-2
= 2
2
2
-2
-2
The slope of a line passing through the points ( x1 , y1 ) and ( x 2 , y 2 ) is
y − y1
m= 2
x 2 − x1
Find the slope of the line passing through the points and tell whether it rises, falls, is horizontal,
or vertical.
⎛1
⎞ ⎛3
⎞
a.) (4, 2) (-18, 1)
b.) (-7, 3) (-2, 3)
c.) ⎜ , −1⎟ ⎜ , −2 ⎟
⎝5
⎠ ⎝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)
y (dollars)
1
2
4
6
Connect the points with a line. What does the slope of the line represent?
8
10
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
of 3%, it rises 3 feet for every 100 feet of horizontal distance.
grade
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-mobilesubscribers-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.
2
f (x) = − x 2 − x + 5; f (6)
3
⎛ 1⎞
f (x) = −3 + 4 x; f −
⎝ 2⎠
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)
7
5
(0, ) and ( 2, )
2
2
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 = −
1
3
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=
2
x−3
3
2
1
y=− x+
5
2
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:___________
y = 3x
x-intercept:_______
y=7
x = −4
Graph then write an equation of the line that passes through the point ( -3, 4) and m =
2
3
A.) Slope-Intercept Form: y = mx + b
B.) Point-Slope Form: y − y1 = m( x − x1 )
C.) Standard Form: Ax + By = C
1) Write the equation of a line that passes through the point (2,3) and a slope of 2) Write the equation of the ling that passes through the point (7,-4) and m =
−1
2
2
5
Given Two Points: Write the equation of the line that passes through (−2,−1) and (3,4)
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 (−3,3) and (3,−1)
Line 2: through (−2,−3) and (2,3)
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 y = −3 x + 2
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
$12,000
$28,000
year
1994
2006
Thousands
Of $
60
55
50
45
40
35
30
25
20
15
10
years since
1990
5
5
-5
10
15
20
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.
Length, x
(inches)
Price, y
(dollars)
16
288
14 Karat Gold Chains (1 gram per inch)
18
20
24
324
360
432
30
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=−
y = 5x + 1
5
x +2
4
y=
3
1
x−
4
4
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: _____________
2
3.) Write the equation of the line that passes through the point (7,-­‐4) and has a slope of . 5
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 1
y = − x + 6 . 2
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.)
2.)
3.)
4.)
Which
Which
Which
Which
variable is independent?
variable is dependent?
axis should the independent variable be on?
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 = ___________________
Study
Regents Score
Hours
3
80
5
90
2
75
6
80
7
90
1
50
2
65
7
85
1
40
7
100
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.
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)
Life Expectancy (y)
0
47.3
10
50
20
54.1
30
59.7
40
62.9
50
68.2
60
69.7
70
70.8
80
73.7
90
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
List Length Time
Non-Matching
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 2 x − 3 y ≥ −5
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 x + y ≥ 1
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)
(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)
(2,4)
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:
5 x − 2 y ≤ −4
y < -2
y < 2x
x ≤1
2.6 Homework Check whether the given ordered pairs are solutions of the inequality.
y < −9x + 7;
x ≤ −5;
(3,−8)
(−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 ≥ −
1
y
3
Write the inequality whose graph is shown.
x+y<0
5x > −20
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.
Graph the following functions.
f(-4)=
f(-2)=
f(0)=
f(5)=
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)
2)
3)
Open the Desmos app on your iPad
Enter in the equation y = a x − h + k
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 =
1
, h = -2, and k = -5.
2
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 =
1
− , h = -3, and k = 4.
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_________________________________________
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 y = x .
X
y
-3
-2
-1
0
1
2
3
2
3
Use the table of values to graph the following equations
1.) y = x + 1
X
y
-3
-2
-1
0
1
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.) y = x + 1
X
y
-4
-3
-2
-1
0
1
2
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.) y = x − 1
X
y
-3
-2
-1
0
1
2
3
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) y = x − 1
x
y
-2
-1
0
1
2
3
4
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) y = − x
x
y
-3
-2
-1
0
1
2
3
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.) y = 3 x
x
y
-3
-2
-1
0
1
2
3
A.) How does this graph
compare to y = x ?
B.) Left slope: ________
Right slope: _______
C.) Vertex: _________
D.) Axis of Symmetry: ______
7.) y =
X
y
1
x
2
-6
-4
-2
0
2
4
6
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
1.
Vertical Shift Graphs
1.
2.
2.
Flipped Graphs
1.
Wider or More Narrow Graphs
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.) y = − x + 2 + 3
A.) Horizontal shift?
B.) Vertical Shift?
C.) Direction opens? Wider or more narrow?
X
y
-5
-4
-3
-2
-1
0
1
Predict the transformation for the following graphs then use a table to verify the graph
by graphing with a table.
2) y = 2 x − 1 + 1
A.) Horizontal shift?
B.) Vertical Shift?
C.) Direction opens? Wider or more narrow
X
y
-2
-1
0
1
2
3
4
Predict the transformation for the following graphs then use a table to verify the graph
by graphing with a table.
3) y =
1
x −2
2
A.) Horizontal shift?
B.) Vertical Shift?
C.) Direction opens? Wider or more narrow?
X
y
-3
-2
-1
0
1
2
3
2.8 Homework Graph the function and describe the translation of the parent function y = x to attain the graph.
y =6x−7
Translation of y = x
______________________
y = − x + 2 + 11
Translation of y = x
____________________________
y = x +9
Translation of y = x
_______________________
y=
1
x−3+4
3
Translation of y = x
_________________________
y = − x − 8 +1
Translation of y = x
_____________________
y=
1
x +6
2
Translation of y = x
_____________________
Write an equation of the graph shown.
1 Algebra II
1.
Chapter 2 Review Sheet
Given g(x) = x 2 + 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.
1
What is the slope of the line y = − x − 7 ?
5
a. 1/5
b. -7
c. -5
d. -1/5
8.
2
4
x+ ?
3
5
c. -4/5
d. -2/3
e. 7
What is the y-intercept of the line y =
a. 2/3
b. 4/5
e. 5/4
9.
1
What is the x-intercept of the line y = − x − 16 ?
4
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?
−2x − 4y = 8
a.
b.
2x + 4y = 8
c.
2x − 4y = −8
d.
−2x − 4y = −8
e.
2x − 4y = 8
4
2
5
–2
–4
–6
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.
b.
y = 4x + 11
c.
y = 4x + 5
y = −4x + 5
y = 4x − 11
d.
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?
1
15
1
13
a.
b.
c.
y = 2x + 5
y= x+
y= x+
2
2
2
2
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
1
line y = x + 4?
3
a.
b.
c.
y = 3x + 15
y = 3x − 15
y = −3x + 15
1
1
d.
e.
y= x−5
y = x −1
3
3
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.
d.
18.
19.
20.
b.
e.
c.
y = −0.051x + 4.8
y = −0.0051x + 4.81
y = 5.1x + .481
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)
Which of the ordered pairs is not a solution of x < −y + 3?
a. (-3, -5)
b. (0, 2)
c. (-2, 0)
e. (-2, -5)
d. (2, 2)
e. (-1, 3)
⎧⎪ 4x + 1, x < −1
Given f (x) = ⎨
, what is f (−1)?
⎩⎪ 2x − 3, x ≥ −1
a.
21.
y = 0.51x + 4.81
y = −0.51x + 4.81
-5
5
b.
3
c.
-1
⎧ −5, −3 < x < 0
⎪
0 ≤ x ≤ 1 , what is f (1)?
Given f (x) = ⎨ 3,
⎪ 7, 1 < x < 4
⎩
a.
-5
b.
1
c.
0
3
d.
-3
e.
d.
7
e.
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
8
6
4
2
–5
23.
24.
25.
5
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)
⎧ −x, −1 ≤ x ≤ 0
⎪
⎛ 1⎞
Given f (x) = ⎨ x,
0 < x ≤ 1 , what is f ⎜ ⎟ ?
⎝ 2⎠
⎪ −2x, 1 < x ≤ 2
⎩
a.
-1
b.
2
c.
½
-1/2
d.
-2
e.
Which function is represented by the graph shown?
a.
b.
c.
d.
y = 3x + 2
y = 2 − 3x
y = − 3x + 2
y = −3x + 2
e.
y = 3x + 2
4
2
–5
5
–2
–4
26.
What is the equation of the line passing through the point (-1, 3) with a slope of 1/2?
1
1
a.
b.
(y − 3) = (x + 1)
(y + 3) = (x + 1)
2
2
1
1
c.
d.
e.
none of the above
(y − 3) = (x − 1)
(y + 3) = (x − 1)
2
2
27.
28.
Determine the Standard Form of the equation y = −2x + 3 .
(y − 3) = −2(x + 0)
(y + 3) = −2(x + 0)
a.
b.
1
−2x + y = 3
d.
e.
y= x+3
2
Which equation form does NOT describe a linear function?
a.
y = mx + b
Ax + By = C
b.
c.
y= x−h +k
d.
(y − y1 ) = m(x − x1 )
c.
2x + y = 3