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Unit 4,5,7 Study PowerPoint
 In this part of the unit
students will…
•identify the key
distinctions between
functions and
relations in multiple
contexts and
representations;
 • classify situations
as functions or
relations, and both
use and show
multiple
representations of
functions and
relations;
Relations
A relation is a set of ordered pairs.
{(2,3), (-1,5), (4,-2), (9,9), (0,-6), (2,5)}
This is a relation
The domain is the set of all x values in the relation
The domain for the set of ordered pairs
listed above: = {-1,0,2,4,9}
The range is the set of all y values in the
relation
The range for the set of ordered pairs
listed above: ={-6,-2,3,5,9}
Relations
Example of a relation: {(1,6), (2,2), (3,4), (4,8), (5,10) (1,3)}
1
2
3
2
4
3
5
6
8
10
Domain
Range
4
Functions
Functions are one of the most important ideas in mathematics.
A function is a relation every ordered pair has unique x value, or
equivalently, no two ordered pairs have the same x - coordinate.
In general, a function is a set of rules for taking input and producing
unique output.
Examples of Functions:
1.) {(2,4), (3,9), (4,16), (0,0), (5,25)}
2.) All ordered pairs of the form: (person name, social security number)
3. ) If you work by the hour, your pay P is a function of the number of
hours h you work
4.) All ordered pairs of the form: (Georgia citizens, year of birth)
Equivalent Definitions of Functions
A set of ordered pairs in which no two different ordered pairs have
the same first coordinate.
A rule that assigns to each element x in a set A (domain) exactly one
element in set B (range).
A function is a relationship or correlation between a set of input and
a set of output. Each element of the input set (domain)
corresponds to one and only one element of the output set (range).
For example, each person has one and only one birthday.
If we list the students in this class in the domain, and all the days of the
year in the range, the function rule "has a birthday" will match each student
with their birthday. There may be many days of the year left over, but each
student will correspond to one and only one birthday.
Ways to Represent a Function
Symbolic
x,y y  2x
or
y  2x
• Numeric X Y
1
2
5 10
-1 -2
3
• Graphical
6
• Verbal
The y value is two
times the value.
Equation(symbolic) connection
A linear equation is an
equation with solutions that lie
in a straight line when
graphed.
Ordered pairs can be used to
write a solution for a twovariable equation.
Representation explanations
Numeric: tables of points are the most
common way of representing a function
Verbal: Describing the relation in words.
Graphically: a picture that shows the
relationship between the input value and
the output value
Symbolically: an equation that relates to
variables
Example 4:
Mapping
Functions
Example 1: Standard
0
3
Example 3:
Table
1
1
x
0
1
2
-1
2
-1
-1
5
2x + y = 3
Example 2: Slope
intercept
y=-2x + 3
y
3
1
-1
5
Vertical Line Test for Functions
Since a function cannot have two different ordered pairs with the
same first component, we can tell if a relation is a function by
looking at the graph.
If every vertical line intersects the
graph of a relation in no more than one point, then the
relation represents a functions
Vertical Line Test for Functions
y
y
x
Function
x
Function
y
x
Not a Function
y
x
Not a function
y
y
x
Function
x
Function
Examples
Example 1:
Does the following set of ordered pairs represent a function ?
{ (2,5), (12, -4), (-4, 12), (1,5) (0,0) (34, 59) } Yes or no
Example 2: Is the following relation a function ?
Yes or no
x
y
0
3
1
1
2
-1
0
5
Function Notation
The Symbolic Form
A truly excellent notation. It is concise and
useful.
y  f x 
y  f x 
Name of the
function
• Output Value
• Member of the Range
• Dependent Variable
• Input Value
• Member of the Domain
• Independent Variable
These are all equivalent names for
the y.
These are all equivalent names for
the x.
Function Notation
We commonly call functions by letters. Because
function starts with f, it is a commonly used letter to
refer to functions.
f  x   2 x  3x  6
2
This means the
right hand side
is a function
called f
This means the
right hand side has
the variable x in it
The left side DOES NOT MEAN
f times x like parenthesis usually
do, it simply tells us what is on
the right hand side.
•The left hand side of this equation is the function notation. It tells us two things:
we called the function f and the variable in the function is x.
•Find f(3)
•f(3) = 2(3)2 – 3(3) + 6
•
=2(9) -9 + 6
•
=18-9 + 6
•
=15
Examples
For each function find f(2) and f(-1):
a.) f(x) = x + 7
f(2) =
f(-1)=
b.) f(x) = x2 + 7
c.) f(x) = 1/(x+1)
f(2)
f(-1)
f(2)=
f(-1)=
Real Life Application of Functions
You are driving a car 50 miles per hour
(mph)
D=rt
d = 50t
d=distance
R= rate t=time
Function
Set up a table using the previous
function rule: d = 50 t
Time
(Input)
Distance
(Output)
1
2
3
4
5
Function
Find the Rule ~ Remember to define
the variable. Look at next slide to see if you
are correct
Hours
Worked(h)
1
2
3
4
5
Salary(s)
$6.50
$13.00
$19.50
$26.00
$32.50
Answer
Define the variable
s = salary
h = hours
Function Rule
Plant Growth Function
Table(Numeric) Representation
Weeks(x)
Plant Height(y)
1
2
3
4
5
3
5
7
9
11
Rabbit Growth Function
Symbolic and Verbal Form
 Equation
In words
Plot points on grid to see the graphical representation
Is this an increasing graph or a decreasing graph?
?
Homework/Class work Assignments
 Textbook pg. 121, 816 all
 Triangle /Perimeter
Multiple
Representation Task
 Problem Solving 3-4
Homework
 Practice 3-1 B
 Coach Book, pg. 149
 Shaded/Un shaded
multiple
representation task
 Practice 3-4
Interpreting Graphs and Tables
Exploration 3-3(opener)
Homework
 Text book pg. 128
 1,2,5,6
 Pg. 131: 12
Class work/quiz
 Problem solving 3-3
 Practice 3-3
Part Two:
Introduction to different forms of
Linear Equations and slope
Standards:
 Graph equations of
 Interpret slope as a
the form ax+by = c.
rate of change.
 Translate among
 Determine the
verbal, tabular,
equation of a line
graphic, and algebraic
given a graph,
representations of
numerical information
functions
that defines the line,
or a context involving
a linear relationship
Standards continued…
 Graph equations of
the form y=mx + b
 Determine the
meaning of slope and
y-intercept in a given
situation
 Determine the
meaning of x and y
intercept
 Vocabulary: Key
components of a
graph
1.Ordered pairs
2.X intercept
3.Y intercept
4.slope
Ordered pairs/coordinates (x , y)
 If the ordered pair lie on the line then it is a solution to
the equation.
 Substitute the ordered pair into the equation and it will
make a true statement if it is a solution
Tell whether the order pair is a solution of
2x-y =5
A. (1,-3)
B. (4,7)
Graphs
The graph of an equation in two variables
is the set of points in a coordinate plane
that represent all solutions of the equation.
An equation whose graph is a line is called
a linear equation.
Pictures of graph




Graph 1:
y=x + 2
x-y=-2
You will learn how to
create an equation from
the graph y using the key
components.
 Graph 2:
 y=3x
 -3x + y =0
Why do we study these
graphs…Applications in real life
Beth begins with $10 in the bank and
saves $5 each week.
 How can we determine how much money
she will have at the end of 1 month? 1
Year?
A plumber charges $38 to travel to your
home for repairs and $100 per hour.
How much will it cost you if he works 5
hours?
X intercept
The x intercept of a graph is the point
where the graph crosses the x axis. It is
also thought of as the place on the graph
where y is equal to zero.
To write the x intercept as an ordered pair:
(x coordinate,0)
Y intercept
The y-intercept of a graph is the point
where the graph crosses the y axis. It is
also considered the place where x is equal
to zero?
(0, y intercept)
Interpreting Graphs…
Graphs and tables can show important
information if you know how to read them.
Tables are good for numbers and finding
specific information quickly, but sometimes
they leave out information.
Graphs give you a visual idea of changes
in data, but are not as specific or detailed
as the other ways.
Slope in Graphs… Why is it important?
To help us determine the relationship
between two quantities.
Slope Applications
Slope Applications
Slope: Multiple Meanings….
Slope tells a lot about the graph.
It is the relationship between the input and
the output
The measure of the steepness of a line on
a graph
Rise/ run
Slope continued…
 Rate of change
 The slope between any two points is always constant in
a linear relationship
 Fraction or ratio
 The difference in the vertical change divided by the
difference in the horizontal change
 The change in y values divided by the change in x
values
4 types of slopes
(direction of the lines)
Positive (x and y moving in the same
direction; increasing)
Negative (x and y moving in the same
direction; decreasing)
Horizontal( the value of x is changing while
y is constant)
Vertical (the value of y is changing while x
is constant )
Types of Lines
created by these
linear
relationships
Positive
Slope
Lines that have positive
slope, slant "up hill"
(as viewed from left to
right).
SkiBird has to work hard
to make it up the hill. He
needs to use positive (+)
energy to get up the hill.
Negative Slope
Lines that have negative
slope, slant "down hill"
(as viewed from left to
right).
SkiBird enjoys the ride
down the hill. He needs
to occasionally use
negative
(-) energy to try to slow
down
No Slope or
Slope Undefined
Vertical lines have no
slope, or undefined
slope.
SkiBird cannot ski
vertically. Sheer doom
awaits SkiBird at the
bottom of a vertical hill.
Zero
Slope
Lines that are horizontal
have zero slope.
SkiBird is cross-country
skiing on level
ground. He is not
working hard to get up a
hill, nor is he trying to
slow down. His energy
level (and his enjoyment
level) is at zero
How to calculate slope? You will learn to
calculate slope from……..
Tables
Graphs
Problem scenarios
equations
FINDING THE SLOPE OF A L INE
The slope m of the non-vertical line
passing through the points (x1, y1)
y
and (x2, y2) is
(x2, y2)
(x1, y1)
Read y1 as “y sub one”
Read x1 as “x sub one”
(y2 - y1 )
(x2 - x1 )
x
m =
rise
run
=
change in y
change in x
= xy22 -- yx11
FINDING THE SLOPE OF A L INE
When you use the formula for the slope,
m
y2 - y1
m =
x1
y -y
rise
changexin
2 -y
= run = change in x = x2 - x1
2
1
The order of subtraction
important. You
label either point as (x1, y1)
theis numerator
andcan
denominator
and the other point as
(x2,use
y2).the
However,
both the numerator
numerator
denominator
must
same subtraction
order. and denominator
must use the same order.
CORRECT
y2 - y 1
x2 - x1
Subtraction order is the same
INCORRECT
y2 - y1
x1 - x2
Subtraction order is different
Find the slope of the line below. Choose
Two points
y from the graph.
Slope (m) = rise
run
Slope (m) = +1
+3
+3
+1
x
m= 1
3
Algebraic Method to Find Slope Given Two Points
1
2
Find the slope between (-1,3) and (4,7).
Slope (m) =
y
x =
y2- y 1
x2- x 1
7-3
Slope (m) =
4 - -1
4
Slope (m) =
5
m=
4
5
Algebraic Method to Find Slope Given Two Points
1
2
Find the slope between (6,5) and (-1,7).
y2 - y1
m = x-x
2
1
7-5
m =
-1 - 6
2
m =
-7
m =
-2
7
A Line with a Zero Slope is Horizontal
(-1, 2)
2) and (3,
Find the slope of a line passing through (-1,
(3, 2).
2).
y
SOLUTION
9
(x11,, yy11)) = (-1, 2) and
Let (x
(x22, y2) = (3,
(3, 2)
2)
7
m=
y2 - y1
x2 - x1
2
= 32- -(-1)
=
0
4
= 0
Rise: difference of y-values
5
rise = 2 - 2 = 0 units
Run: difference of x-values
(-1, 2)
3
(3, 2)
Substitute values.
run = 3 - ( -1) = 4 units
1
Simplify.
-1
-1
1
3
Slope is zero. Line is horizontal.
5
7
9
x
INTERPRETING SLOPE AS A RATE OF CHANGE
Slope as a Rate of Change
You are parachuting. At time tt == 00 seconds,
2500 feet
feet
seconds,you open your parachute at hh == 2500
above the ground. At t = 35 seconds,
seconds, you are at h = 2115 feet.
y
2700
a. What is your rate of change in height?
(0, 2500)
2500
b. About when will you reach the ground?
Height (feet)
2300
2100
(35, 2115)
1900
5
15
25
Time (seconds)
35
45 x
Slope as a Rate of Change
SOLUTION
a. Use the formula for slope to find the rate of change.
change. The change
change in time is
35 - 0 = 35 seconds. Subtract in the same order. The change
change in height
height is
2115 - 2500 = -385 feet.
VERBAL
MODEL
LABELS
Rate of
Change
=
Change in Height
Change in Time
Rate of Change =
m (ft/sec)
Change in Height = -- 385 (ft)
Change in Time =
ALGEBRAIC
MODEL
m =
35 (sec)
- 385
35
= -11
Your rate of change is -11 ft/sec. The negative value indicates you are falling.
Different forms of Linear Equations
 3x + y = 7
 Standard form
 To graph an equation
in standard form we
use the intercepts
 Solve the equation for
y to change to slope
intercept form
 y=-3x +7
 Slope-intercept form
 To graph an equation
in slope intercept form
we use the slope and
the y intercept.
STANDARD FORM
Standard form of a linear equation is Ax + By = C. A and B are
not both zero. A quick way to graph this form is to plot its
intercepts (when they exist).
Draw a line through the two points.
y
((x(x, ,0)0)
x
Ax + By = C
STANDARD FORM
GRAPHING EQUATIONS IN STANDARD FORM
The standard form of an equation gives you a
quick
way to graph the equation.
1 Write equation in standard form.
2 Find x-intercept by letting y = 0. Solve for x. Use
x-intercept to plot point where line crosses x-axis.
3 Find y-intercept by letting x = 0. Solve for y. Use
y-intercept to plot point where line crosses y-axis.
4 Draw line through points.
Drawing Quick Graphs
Graph 2x + 3y = 12
(0, 4)
SOLUTION
METHOD 1: USE STANDARD FORM
2x + 3y = 12
2x + 3(0) = 12
Standard form.
Let y = 0.
x=6
Solve for
x. plot the point (6, 0).
The x-intercept is 6, so
2(0) + 3y = 12
Let x = 0.
y=4
Solve for
y. plot the point (0, 4).
The y-intercept is 4, so
Draw a line through the two points.
(6, 0)
Standard Form Problems: Graph using x
and y intercepts
1.
2.
3.
4.
5.
6.
7.
2x – 4y =8
3x + 4y =12
4x + 2y = 6
2x + 8y = 16
3x + 9y = 6
x- 6y = 18
x-3y = 3
Objective
To be able to recognize
Horizontal and Vertical
lines on the coordinate
plane.
Plot the points
(3,-3), (3,-2), (3,-1), (3,0),(3,1), (3,2),
(3,3)
What do you
notice about
all the x
values that
were plotted?
The
Equation
of the
line is x
=3
Now you Plot the points
(1,-3), (1,-2), (1,-1), (1,0),(1,1), (1,2),
(1,3)
What is
the
equation
of the
line?
The
Equation
of the
line is x
=1
What did you notice about all
the x values?
When the x values are
the same, what does the
equation begin with?
Plot the points
(-2,2), (0,2), (3,2)
What is the equation
of the line?
What did you notice about all the
y values?
When the y values are
the same, what does the
equation begin with?
STANDARD FORM
The equation of a vertical line cannot be written in slopeintercept form because the slope of a vertical line is not defined.
Every
linear equation, however, can be written in standard form—
even the equation of a vertical line.
HORIZONTAL AND VERTICAL LINES
HORIZONTAL LINES The graph of y =
through (0, c ).
VERTICAL LINES
c is a horizontal line
The graph of x = c is a vertical line
through (c , 0).
Graphing Horizontal and Vertical Lines
Graph y = 3 and x = –2
y=3
SOLUTION
The graph of y = 3 is a horizontal
line that passes through the point
(0, 3). Notice that every point on the
line has a y-coordinate of 3.
The graph of x = –2 is a vertical line
that passes through the point (– 2, 0).
Notice that every point on the line has
an
x-coordinate of –2.
(0, 3)
x = –2
(–2, 0)
SLOPE-INTERCEPT FORM
If the graph of an equation intersects the y -axis at the point
(0, b), then the number b is the y -intercept of the graph. To
find the y -intercept of a line, let x = 0 in an equation for the
line and solve for y.
The slope intercept form of a
linear equation is
y = mx + b.
m
b
y
(0 , b)
x
is the slope
y
is the -intercept
y = mx + b
SLOPE-INTERCEPT FORM
GRAPHING EQUATIONS IN SLOPE-INTERCEPT FORM
The slope-intercept form of an equation gives you a quick
way to graph the equation.
Write equation in slope-intercept form by solving
for y.
STEP 2 Find y-intercept, use it to plot point where line
crosses
y-axis.
STEP 3 Find slope, use it to plot a second point on
line.
STEP 4 Draw line through
points.
STEP 1
Graphing with the Slope-Intercept Form
Graph y =
3
x–2
4
(4, 1)
SOLUTION
The equation is already in slopeintercept form.
3
(0, – 2)
4
The y-intercept is – 2, so plot the
point (0,
(0,– –2)2)where the line crosses the
y -axis.
3
The slope is 4 , so plot a second point on the line by
moving
(4, 1).
4 units to the right and 3 units up. This point is (4, 1).
Draw a line through the two points.
Standard Form Problems: Solve each for
“y” and graph using the slope-intercept
method.
1.
2.
3.
4.
5.
6.
7.
2x – 4y =8
3x + 4y =12
4x + 2y = 6
2x + 8y = 16
3x + 9y = 6
x- 6y = 18
x-3y = 3
Using the Slope-Intercept Form
In a real-life context the y-intercept often represents an initial
amount and the slope often represents a rate of change.
You are buying an $1100 computer on layaway. You make
a $250 deposit and then make weekly payments according
to the equation a = 850 – 50 t where a is the amount you
owe and t is the number of weeks.
What is the original amount
you owe on layaway?
What is your weekly payment?
Graph the model.
Using the Slope-Intercept Form
What is the original amount you owe on layaway?
SOLUTION
First rewrite the equation asa a==– –5050t t++850
850 so that it is
in
slope-intercept form.
Then you can see that the a-intercept is
850.
So, the original amount you owe on layaway
(the amount when t = 0) is $850.
Using the Slope-Intercept Form
a = – 50tt++850
850
What is your weekly payment?
SOLUTION
From the slope-intercept form you can see that
the slope is m = – 50.
This means that the amount you owe is changing at
a rate of – 50 per week.
In other words, your weekly payment is $50.
Using the Slope-Intercept Form
a = – 50 t + 850
Graph the model.
(0, 850)
SOLUTION
Notice that the line stops when
it reaches the t-axis
= 17).
(at (at
t = t17).
The computer is completely
paid
for at that point.
(17, 0)