Transcript Functions and Their Graphs 2.1 1 2

2.1 Functions and Their Graphs
What you should learn:
Goal 1 Represent relations and functions.
Goal 2 Graph and evaluate linear functions.
2.1 Functions and Their Graphs
graph
y  2x 1
construct a table of values
x
y  2x 1
-2
-3
-1
0
1
2
-1
1
3
5
Now, let’s write ordered pairs, and graph.
The ordered pairs are
(-2, -3), (-1, -1), (0, 1), (1, 3), 2, 5)
Plot these points on a coordinate plane.
And draw the line.
Graph the linear equation. Select integers for x,
starting with -2 and ending with 2. Organize you
work in a table.
2
graph
y  x 1
3
construct a table of values
x
2
y 
x 1
3
-2
-1
7

3
5

3
0
1
2
1
1 
3
1
3
Now, let’s write ordered pairs, and graph.
The ordered pairs are
7 
5

 1  1
  2, ,   1, , 0,1, 1, ,  2, 
3 
3

 3  3
Plot these points on a coordinate plane.
And draw the line.
Plot these ordered pairs
7 
5

 1  1



2
,

,

1
,

,
0
,

1
,



1, ,  2, 
3 
3

 3  3
y  x 4
Graph the equation:
2
Notice that because the x is squared, the graph is not a
linear equation.
To graph:
1. Make a table.
2. Use 7 points.
3. Find ordered pairs.
x
3
2
1
0
-1
-2
-3
y  x2  4
5
0
-3
-4
-3
0
5
(3, 5)
(2, 0)
(1, -3)
(0, -4)
(-1, -3)
(-2, 0)
(-3, 5)
Plot these ordered pairs
(3, 5)
(2, 0)
(1, -3)
(0, -4)
(-1, -3)
(-2, 0)
(-3, 5)
Reflection on the Section
When is a relation a function?
assignment
2.1 Functions and Their Graphs
2.2 Slope and Rate of Change
What you should learn:
Goal 1 Find slopes of lines and classify
parallel and perpendicular lines.
Goal 2 Use slope to solve real-life problems.
2.2 Slope and Rate of Change
The slope of the non-vertical line passing
through the points x y  and x y  is
1,
2,
1
2
y
y
m
x x
2
1
2
1
The numerator is read as “ y sub 2 minus y sub 1”
and is called the rise.
The denominator is read as
and is called the run.
“ x sub 2 minus x sub 1”
Find the Slope of the line passing through each pair of points or
state that the Slope is undefined.
ex1) (5,
6) and (-3, 2)
Make the substitution.
26
m
35
Do the math.
4
m
8
1

2
use
y
y
m
x x
2
1
2
1
Find the Slope of the line passing through each pair of points or
state that the Slope is undefined.
ex2) (1,
-4) and (-2, -4)
Make the substitution.
 4  (4)
m
 2 1
Do the math.
0
m
3
0
use
y
y
m
x x
2
1
2
1
Find the Slope of the line passing through each pair of points or
state that the Slope is undefined.
ex3) (9,
5) and (9, 1)
use
Make the substitution.
1 5
m
99
Do the math.
4
m
0
 undefined
y
y
m
x x
2
1
2
1
Find the slope given 2 points.
ex)
( 1998, 1502),
(2004, 1112)
1112  1502
m
2004  1998
 390
m
 65
6
use the formula…
y
y
m
x x
2
1
2
1
this is your slope
Classification of lines by Slope
1. A line with positive slope rises from
left to right. (m > 0)
2. A line with negative slope falls from
left to right. (m < 0)
3. A line with slope zero is horizontal.
(m = 0)
4. A line with undefined slope is vertical.
(m is undefined)
directions
Find the slope of each line, or
state that the slope is undefined.
Count the Rise 7
Count the Run
8
So, the SLOPE is
7
8
or also written
7
m
8
directions
Find the slope of each line, or
state that the slope is undefined.
Count the Rise 6
Count the Run
8
So, the SLOPE is
6

8
3
or reduced to m  
4
directions
Find the slope of each line, or
state that the slope is undefined.
Count the Rise 7
Count the Run 0
So, the SLOPE is
7
0
This fraction is undefined.
So, m is undefined.
directions
Find the slope of each line, or
state that the slope is undefined.
Count the Rise 0
Count the Run 6
So, the SLOPE is
0
6
This fraction is zero.
So, m = 0.
Slope and Parallel Lines
1. If two non-vertical lines are parallel, then they have
the same slope.
2. If two distinct non-vertical lines have the same slope,
then they are parallel.
3. Two distinct vertical lines, each with undefined slope,
are parallel.
4. Because two parallel have the same “steepness”, they
must have the same slope.
Slope and Perpendicular Lines
1. If two non-vertical lines are perpendicular, then the
product of their slopes is -1.
2. If the product of the slopes of two lines is -1, then the
lines are perpendicular.
3. A horizontal line having zero slope is perpendicular to
a vertical line having undefined slope.
Determine whether the lines through each pair of points are parallel.
ex)
(3, 8) and (-5, 4)
(4, 2) and (8, 4)
42
m
84
48
m
53
4
m
8
1

2
2
m
4
Same slopes,… parallel.
1

2
Determine whether the lines through each pair of points are parallel.
ex)
(-4, 2) and (3, 0)
(-2, 5) and (0, 12)
02
m
3  (4)
2
m
7
12  5
m
0  (2)
7
m
2
perpendicular
Using Slope in Real Life
In a home repair manual the following ladder safety guideline is given.
Adjust the ladder until the distance from the base of the ladder to the wall is
at least one quarter of the height where the top of the ladder hits the wall.
Find the minimum distance a ladder’s base should be from a wall if you need
the ladder to reach a height of 20 feet.
SOLUTION
4
Letrise
x represent
the minimum
distance
that the ladder’s
Write
a proportion.
=
1 be from the wall for the ladder to safely
runshould
base
reach
of 20 feet.
20 a height
4
The rise is 20 and the run is x.
x =1
20 = 4x
5=x
Cross multiply.
Solve for x.
The ladder’s base should be at least 5 feet from the wall.
Finding the Slope of a Line
In real-life problems slope is often used to describe an average rate of
change. These rates involve units of measure, such as miles per hour or
dollars per year.
DESERTS In the Mojave Desert in California, temperatures can drop quickly
from day to night. Suppose the temperature drops from 100ºF at 2 P.M. to
68ºF at 5 A.M. Find the average rate of change and use it to determine the
temperature at 10 P.M.
SOLUTION
Average rate of change =
=
Change in temperature
Change in time
–32ºF
68ºF – 100ºF
 –2ºF per hour
=
15 hours
5 A.M. – 2 P.M.
Because 10 P.M. is 8 hours after 2 P.M., the temperature changed 8(–2ºF) = –16ºF.
That means the temperature at 10 P.M. was about 100ºF – 16ºF = 84ºF.
Reflection on the Section
How can you tell from a line’s graph if it has
positive, negative, or zero slope?
assignment
2.2 Slope and Rate of Change
2.3 Quick Graphs of Linear Equations
What you should learn:
Goal 1 Use the slope-intercept form of a
linear equation to graph linear
equations.
Goal 2 Use the standard form of a linear
equation to graph linear equations.
2.3 Quick Graphs of Linear Equations
Intercepts of a line
Using Intercepts to Graph Ax + By = C.
(this is the Standard Form of a Linear equation.)
1. To find the x-intercept, let y = 0 and solve for x in Ax = C.
2. To find the y-intercept, let x = 0 and solve for y in By = C.
3. Find a checkpoint, a third ordered-pair.
4. Graph the equation by drawing a line through
the three points.
Finding the x- and y- intercepts….
Find the x-intercept of the equation
2x + 3y = 6.
To find the x-intercept, substitute (0) in for y.
Solve for x.
2x + 3(0) = 6
2x = 6
x=3
The
coordinate
So, that means the x-intercept is 3 or (3, 0)
Find the y-intercept of 2x + 3y = 6.
To find the y-intercept, substitute (0) in for x.
Solve for y.
2(0) + 3y = 6
3y = 6
y=2
The
coordinate
So, that means the y-intercept is 2 or (0, 2)
Plot these ordered pairs
x-int (3, 0)
y-int (0, 2)
Know if you
connect the dots,
this is the line
representing:
2x + 3y =6
Find the x- and y- intercepts of each equation.
Do not graph, yet.
ex) -x + 4y = 8.
To find the y-intercept,
substitute (0) in for x.
Solve for y.
To find the x-intercept,
substitute (0) in for y.
Solve for x.
-(0) + 4y = 8
-x + 4(0) = 8
-x = 8
The
coordinates
x = -8
(-8, 0)
( 0, 2)
4y = 8
y=2
Use the x and y intercepts and a check point to graph
each equation.
Ex)
4 x  5 y  20
x-intercept
4x + 5(0) = 20
4(1)  5 y  20
y  3.2
y-intercept
4(0) + 5y = 20
4x = 20
5y = 20
x=5
y=4
That’s the
coordinate
( 1, 3.2)
Pick x = 1
x-intercept
x- axis
y-intercept
y- axis
Use a checkpoint, to see if the line is in the right spot. Do this
by picking an x-coordinate, substitute, and solve for y.
Write an equation in slope-intercept form of
the line that passes through the two points.
Ex )
x-intercept = 4
y-intercept = 2
What
coordinates
are these?
1. Find the slope
2. You have a yint. and a slope.
3. write in slopeintercept form.
Horizontal and Vertical Lines
The graph of a linear equation in one variable is a
horizontal or vertical line.
The graph of y = b is a horizontal line.
The graph of x = a is a vertical line.
Draw the graph and write an equation for the
horizontal line that passes through the point (-2,3).
The equation for this
line is…
y3
Why is this?
Because if you go to any
point on this line, the
(y) coordinate of the
ordered pair ( x, y )
would be 3.
( ?, 3 ) always
( 4, 3)
Draw the graph and write an equation for the
vertical line that passes through the point (-2,3).
The equation for this
line is…
x  2
Why is this?
Because if you go to any
point on this line, the
(x) coordinate of the
ordered pair ( x, y )
would be -2.
( -2, ? ) always
( -2, -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
50 t++850
850 so that it is in
First rewrite the equation as aa == –– 50t
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
50tt++850
a = – 50
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 (at t = 17).
The computer is completely paid
for at that point.
(17, 0)
Reflection on the Section
Give an advantage of graphing a line using the
slope-intercept form of its equation.
assignment
2.3 Quick Graphs of Linear Equations
2.4 Writing Equations of Lines
What you should learn:
Goal 1 Write linear equations.
Goal 2 Write direct variation equations.
2.4 Writing Equations of Lines
Summary of Equations of Lines
Slope of a line through
two points:
Vertical line
Horizontal line
Slope-Intercept form:
Point-Slope form:
Standard form:
y2  y1
m
x2  x1
x=a
y=b
y  mx  b
y  y1  m( x  x1 )
Ax  By  C
Point-Slope Form
y  y1  m( x  x1 )
This means that if you have a slope and a
point, you will now use this formula to write
the equation for the line.
How about some examples…?
Write an equation of the line that passes through the
point and has the given slope.
Then write in Slope-Intercept Form.
Ex 1)
(2, 3) , m = 2
y  y1  m( x  x1 )
y  3  2( x  2)
substitute
put in Slope-Intercept form
y  2x 1
Write an equation in slope-intercept form of
the line that passes through the two points.
Ex 2)
( 3, 1), ( -5, 9)
Oh my,
what do I
do?
1. Find the slope
2. You have a
point and a
slope.
3. Rewrite in
slope-intercept
form.
Reflection on the Section
Describe how to determine that a set of data values
exhibits direct variation, and how to find the
constant of variation.
assignment
2.4 Writing Equations of Lines
2.5 Correlation and Best-Fitting Lines
What you should learn:
Goal 1 Use a scatter plot to identify the
correlation shown by a set of data.
Goal 2 Approximate the best-fitting line for a
set of data.
2.5 Correlation and Best-Fitting Lines
Guidelines: to visualize the relationship
between two variables:
Write each pair of values as an ordered
pair (x , y).
In a coordinate plane, plot points that
correspond to ordered pairs.
Use the scatter plot to describe the
relationship between the variables.
m (mph) 0 5 10 15 20 25 30 35 40
f (ft/sec) 0 7 14 22 29 36 44 51 58
feet per second
60
50
40
30
20
10
10 20 30 40 50
Miles per Hour
Write the ordered pairs that correspond to the
points labeled on the coordinate plane.
A(-8,5)
E
A
C
B(-6,-5)
C(0,5)
D(4,0)
D
E(6,7)
B
F
F(8,-5)
Reflection on the Section
How do you use the best-fitting line to make a
prediction?
assignment
2.5 Correlation and Best-Fitting Lines
2.6 Linear Inequalities in Two Variables
What you should learn:
Goal 1 Graph linear inequalities in two
variables
Goal 2 Use linear inequalities to solve reallife problems, such as finding the
number of minutes you can call
relatives using a calling card.
2.6 Linear Inequalities in Two Variables
Sketching the Graph of a Linear Inequality
1.Sketch the graph of the corresponding
linear equation. (Use a dashed line for
inequalities with < or > and a solid line
for inequalities with  or  ) This line
separates the coordinate plane into two
half planes.
example) Sketch the graph.
2x  y  1
You are going to sketch the graph of …
2x  y  1
a.) Solve for y
b.) find x- and y-int
2. Test a point in one of the half planes to
find whether it is a solution of the
inequality.
2x  y  1
We will test the point (0, 0) by substituting
into the original inequality.
2(0)  (0)  1
0 1
TRUE
So,…we shade that side of the plane.
sketch the graph of …
3x  y  3
Graph the line
3x  y  3
either solve for y,
(slope-intercept form)
or find x- and y-int
Test an easy to deal with
point…like (0,0)
3(0)  (0)  3
03
False
So, shade in the
other side.
sketch the graph of …
y4
Graph the line
y4
Test an easy to deal with
point…like (0,0)
04
False
So, shade in the
other side.
Using Linear Inequalities in Real Life
You have relatives living in both the United States and Mexico. You are given
a prepaid phone card worth $50. Calls within the continental United States
cost $.16 per minute and calls to Mexico cost $.44 per minute.
Write a linear inequality in two variables to represent the number of minutes you
can use for calls within the United States and for calls to Mexico.
SOLUTION
Verbal Model
Labels
…
United
States
rate
•
United
States
time
+
Mexico
rate
United States rate = 0.16
United States time = x
Mexico rate = 0.44
Mexico time = y
Value of card = 50
•
Mexico
time

Value of
Card
(dollars per minute)
(minutes)
(dollars per minute)
(minutes)
(dollars)
Using Linear Inequalities in Real Life
You have relatives living in both the United States and Mexico. You are given
a prepaid phone card worth $50. Calls within the continental United States
cost $.16 per minute and calls to Mexico cost $.44 per minute.
Write a linear inequality in two variables to represent the number of minutes you
can use for calls within the United States and for calls to Mexico.
…
Algebraic
Model
0.16 x + 0.44 y  50
Graph the inequality and discuss three possible
solutions in the context of the real-life situation.
Graph the boundary line 0.16 x + 0.44 y = 50 .
Use a solid line because 0.16 x + 0.44 y  50.
Using Linear Inequalities in Real Life
You have relatives living in both the United States and Mexico. You are given
a prepaid phone card worth $50. Calls within the continental United States
cost $.16 per minute and calls to Mexico cost $.44 per minute.
Graph the inequality and discuss three possible solutions in the context of the
real-life situation.
Test
the pointis(0,
Because
(0, 0) is a solution if the
One solution
to0).
spend
65 minutes
inequality,
shade
half-plane
on calls within
thethe
United
Statesbelow
and the line.
Finally,
because
x and
y cannot be negative, restrict
90 minutes
on calls
to Mexico.
the graph to the points in the first quadrant.
Possible
solutions
points
To split time
evenly,are
you
couldwithin
spendthe
83 minutes
shaded
shown.
on
calls region
within the
United States and 83 minutes
on calls to Mexico. The total cost will be $49.80.
You could instead spend 150 minutes on calls
within the United States and only 30 minutes on
calls to Mexico. The total cost will be $37.20.
Reflection on the Section
When is a relation a function?
assignment
2.6 Linear Inequalities in Two Variables
2.7 Piecewise Functions
What you should learn:
Goal 1 Represent piecewise functions
Goal 2 Use piecewise functions to model
real-life quantities.
2.7 Piecewise Functions
Reflection on the Section
A phone company charges in 6 second blocks.
What will a graph of the charges look like?
assignment
2.7 Piecewise Functions
2.8 Absolute Value Functions
What you should learn:
Goal 1 Represent absolute value functions.
Goal 2 Use Absolute Value functions to
model real-life situations.
2.8 Absolute Value Functions
Solve the equation algebraically.
x2 5
solution
x2 5
x7
or
x  2  5
x  3
The equation has two solutions: 7 and -3.
Check these solutions by substituting each
into the original equation.
Check by Sketching the graph of the equation
x2 5
rewrite
x 2 5  y
Find the coordinates of the Vertex.
x-2=0
The coordinates for the
vertex
are
(2,-5)
x=2
Now, make a table. Pick a couple of
x-points less than 2 and a couple of
x-points greater than 2.
x
y
7 0
3 -4
2 -5
1 -4
-3 0
Solutions are the x-intercepts.
Graphs of Absolute Value Equations
How to graph an absolute value
equation.
Sorry…
In this lesson we will learn to sketch the
graph of absolute value. To begin, let’s look
at the graph of y  x .
By constructing a table of values and
plotting points, you can see that the graph is
V-shaped and opens up. The VERTEX of
this graph is (0,0).
x y  x.
-3 3
-2 2
-1 1
0 0
1 1
2 2
y  x.
Vertex at (0,0)
opens up
y   x.
Vertex at (0,0)
opens down
2
y  x2
Vertex at (2,0),
Opens up
y  x 1
Vertex at (0,1),
Opens up
Sketching the graph of an Absolute Value
y  a xb c
1. Find the x-coordinate of the vertex by
finding the value of x for which x + b =0
2. Make a table of values using the xcoordinate of the vertex, some x-values to its
left, and some to its right.
3. Plot the points given, and connect.
Find the coordinates of the vertex of the graph
ex)
y  2 x 1  2
So, what is the value of x when,
x – 1= 0 ?
Yes, 1.
Therefore, the x-coordinate of the vertex is 1.
Now, substitute 1 in for x, then solve for y.
The coordinates are (1,2)
Sketch the graph of the equation
ex)
y  x3 2
Find the coordinates of the Vertex.
x+3=0
The coordinates for the
vertex
are
(-3,-2)
x = -3
Now, make a table. Pick a couple of
x-points less than -3 and a couple of
x-points greater than -3.
x
y
-5 0
-4 -1
-3 -2
-2 -1
-1 0
Reflection on the Section
For the graph of y  a x  h  k
Tell how to find the vertex, the direction the graph
opens, and the slopes of the braches.
assignment
2.8 Absolute Value Functions