Functions and Their Graphs

1
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
2
Key Terms
Relation a mapping of input values with output
values.
Domain input values, (x) Range output values,
(y)
Function a relation is a function provided there
is exactly one output for each input.
3
graphing linear equations
graph
construct a table of values
x -2 -1 0 1 2
-3
-1
1
3
5
Now, lets write ordered pairs, and graph.
4
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.
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Graph the linear equation. Select integers for x,
starting with -2 and ending with 2. Organize you
work in a table.
graph
construct a table of values
x -2 -1 0 1 2
Now, lets write ordered pairs, and graph.
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The ordered pairs are
Plot these points on a coordinate plane. And draw
the line.
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Plot these ordered pairs
9
Graph the equation
Notice that because the x is squared, the graph
is not a linear equation.
To graph
x
(3, 5)
5
1. Make a table.
3
(2, 0)
2
0
2. Use 7 points.
-3
1
(1, -3)
3. Find ordered pairs.
-4
(0, -4)
0
-3
(-1, -3)
-1
-2
(-2, 0)
0
(-3, 5)
-3
5
10
Plot these ordered pairs
(3, 5)
(2, 0)
(1, -3)
(0, -4)
(-1, -3)
(-2, 0)
(-3, 5)
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Vertical Line Test a relation is a function if
and only if no vertical line intersects the graph
of the relation at more than one point.
12
Evaluate the function when x 3.
ex1)
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Reflection on the Section
When is a relation a function?
assignment
PAGE 71 1-50
2.1 Functions and Their Graphs
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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
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The slope of the non-vertical line passing
through the points and
is
The numerator is read as y sub 2 minus y sub 1
and is called the rise.
The denominator is read as x sub 2 minus x sub
1 and is called the run.
2.2 Slope and Rate of Change
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Find the Slope of the line passing through each
pair of points or state that the Slope is
undefined.
(5, 6) and (-3, 2)
ex1)
use
Make the substitution.
Do the math.
2.2 Slope and Rate of Change
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Find the Slope of the line passing through each
pair of points or state that the Slope is
undefined.
(1, -4) and (-2, -4)
ex2)
use
Make the substitution.
Do the math.
2.2 Slope and Rate of Change
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Find the Slope of the line passing through each
pair of points or state that the Slope is
undefined.
(9, 5) and (9, 1)
ex3)
use
Make the substitution.
Do the math.
2.2 Slope and Rate of Change
19
Find the slope given 2 points.
( 1998, 1502), (2004, 1112)
ex)
use the formula
this is your slope
2.2 Slope and Rate of Change
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Classification of lines by Slope
1. A line with positive slope rises from left to
right. (m gt 0)
2. A line with negative slope falls from left to
right. (m lt 0)
3. A line with slope zero is horizontal. (m
0)
4. A line with undefined slope is vertical. (m
is undefined)
2.2 Slope and Rate of Change
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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
or also written
2.2 Slope and Rate of Change
22
directions
Find the slope of each line, or state that the
slope is undefined.
Count the Rise
6
8
Count the Run
So, the SLOPE is
or reduced to
2.2 Slope and Rate of Change
23
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
This fraction is undefined. So, m is undefined.
2.2 Slope and Rate of Change
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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
This fraction is zero. So, m 0.
2.2 Slope and Rate of Change
25

• Slope and Parallel Lines
• If two non-vertical lines are parallel, then they
  have the same slope.
• If two distinct non-vertical lines have the same
  slope, then they are parallel.
• Two distinct vertical lines, each with undefined
  slope, are parallel.
• Because two parallel have the same steepness,
  they must have the same slope.

2.2 Slope and Rate of Change
26

• Slope and Perpendicular Lines
• If two non-vertical lines are perpendicular, then
  the product of their slopes is -1.
• If the product of the slopes of two lines is -1,
  then the lines are perpendicular.
• A horizontal line having zero slope is
  perpendicular to a vertical line having undefined
  slope.

2.2 Slope and Rate of Change
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Determine whether the lines through each pair of
points are parallel or perpendicular.
(3, 8) and (-5, 4) (4, 2) and (8, 4)
ex)
Same slopes, parallel.
2.2 Slope and Rate of Change
28
Determine whether the lines through each pair of
points are parallel or perpendicular.
(-4, 2) and (3, 0) (-2, 5) and (0, 12)
ex)
perpendicular
2.2 Slope and Rate of Change
29
Find the minimum distance a ladders base should
be from a wall if you need the ladder to reach a
height of 20 feet.
SOLUTION
Let x represent the minimum distance that the
ladders base should be from the wall for the
ladder to safely reach a height of 20 feet.
Write a proportion.
The rise is 20 and the run is x.
20 4x
Cross multiply.
5 x
Solve for x.
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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
? 2ºF per hour
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.
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Reflection on the Section
How can you tell from a lines graph if it has
positive, negative, or zero slope?
assignment
Page 79 1-44
2.2 Slope and Rate of Change
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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
33

• Intercepts of a line
• Using Intercepts to Graph Ax By C.
• (this is the Standard Form of a Linear equation.)
• 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.

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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
The coordinate
x 3
So, that means the x-intercept is 3 or (3, 0)
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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)
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Plot these ordered pairs
(3, 0)
x-int
(0, 2)
y-int
Know if you connect the dots, this is the line
representing 2x 3y 6
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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
4y 8
-x 8
The coordinates
y 2
x -8
(-8, 0) ( 0, 2)
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Use the x and y intercepts and a check point to
graph each equation.
Ex)
Thats the coordinate ( 1, 3.2)
x-intercept
y-intercept
4x 5(0) 20
4(0) 5y 20
5y 20
4x 20
y 4
x 5
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.
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Slope-Intercept form of the equation of a line.
y mx b
The slope is m. The y-intercept is b.
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Sketch the graph for the line
Example 1)
To do this find the (Slope) and the (y
intercept).
y-intercept
slope
(0, -8)
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m 4, ( 0, -8)
What you will do to graph is
1. Go to the point (0, -8), put a dot.
2. Move up (rise), 4 spaces , then right (run) 1
space, put a dot.
3. Connect the dots to form a line.
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Sketch the graph for the line
Example 2)
y-intercept
(0, 5)
slope
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( 0, 5)
What you will do to graph is
1. Go to the point (0, 5), put a dot.

• Move down (rise), 3 spaces , then right (run) 4
  space, put a dot.

3. Connect the dots to form a line.
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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.
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Draw the graph and write an equation for the
horizontal line that passes through the point
(-2,3).
The equation for this line is
( 4, 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
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Draw the graph and write an equation for the
vertical line that passes through the point
(-2,3).
The equation for this line is
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)
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In a real-life context the y-intercept often
represents an initialamount 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.
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What is the original amount you owe on layaway?
SOLUTION
First rewrite the equation as a 50t 850
so that it is inslope-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.
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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.
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Graph the model.
SOLUTION
Notice that the line stops when it reaches the
t-axis (at t 17).
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Reflection on the Section
Give an advantage of graphing a line using the
slope-intercept form of its equation.
assignment
Page 86 1-61 odd
2.3 Quick Graphs of Linear Equations
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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
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Summary of Equations of Lines
Slope of a line through two points
x a
Vertical line
y b
Horizontal line
Slope-Intercept form
Point-Slope form
Standard form
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Point-Slope Form
This means that if you have a slope and a point,
you will now use this formula to write the
equation for the line.
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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
substitute
put in Slope-Intercept form
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Write an equation in slope-intercept form of the
line that passes through the two points.
Oh my, what do I do?
( 3, 1), ( -5, 9)
Ex 2)
1. Find the slope
2. You have a point and a slope.
3. Rewrite in slope-intercept form.
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Write an equation of the line that passes through
the point and has the given slope. Then write in
Slope-Intercept Form.
(-2, -1) , m 1
Ex 3)
substitute
put in Slope-Intercept form
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Use the given conditions to write an equation for
each line in point-slope form and slope-intercept
form.
Passing through (2, 3) and parallel to the line
whose equation is
ex)
1. Find the slope of the equation.
2. Use that slope and the new point .
3. Write equation in point-slope form.
4. Rewrite equation in slope-intercept form.
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Parallel slope
and new point (2, 3)
m 4
Point slope form
Slope-intercept form
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Write an equation in Point-Slope form of the line
that passes through the two points.
What coordinates are these?
ex)
x-intercept 4
y-intercept 2
Find the slope
( 4, 0), ( 0, 2)
You have a point and a slope.
Write an equation in POINT-SLOPE form.
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( 4, 0) or ( 0, 2), and
Use EITHER point.
POINT-SLOPE form of an equation
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Reflection on the Section
If you were given the x-intercept of (2), and the
y-intercept of (-4), explain how you would write
an equation in Slope-Intercept Form.
assignment
Page 95 1 - 28
2.4 Writing Equations of Lines
66
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
67

• 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.

68

• Guidelines to correlation
• Positive correlation if y tends to increase as x
  increases.
• Negative correlation if y tends to decrease as x
  increases.
• Relatively no correlation if the points show no
  linear pattern.

69
Approximate the best-fitting line for the data.
Then tell whether x and y have a positive
correlation, a negative correlation, or
relatively no correlation.
I picked
(-5,3) and (3,-4)
70
Approximate the best-fitting line for the data.
Then tell whether x and y have a positive
correlation, a negative correlation, or
relatively no correlation.
So we know
and (-5,3)
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m (mph) 0 5 10 15 20 25 30 35 40
f (ft/sec) 0 7 14 22 29 36 44 51 58
60
50
40
30
feet per second
20
10
10
20
30
40
50
Miles per Hour
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Write the ordered pairs that correspond to the
points labeled on the coordinate plane.
A(-8,5)
B(-6,-5)
E
A
C
C(0,5)
D(4,0)
D
E(6,7)
B
F
F(8,-5)
73
Reflection on the Section
How do you use the best-fitting line to make a
prediction?
assignment
Page 103 1 - 28
2.5 Correlation and Best-Fitting Lines
74
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 real-life
problems, such as finding the number of minutes
you can call relatives using a calling card.
2.6 Linear Inequalities in Two Variables
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Sketching the Graph of a Linear Inequality

• Sketch the graph of the corresponding linear
  equation. (Use a dashed line for inequalities
  with lt or gt and a solid line for inequalities
  with or ) This line separates the
  coordinate plane into two half planes.

Example 1)
Sketch the graph.
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You are going to sketch the graph of
Slope-Intercept Form
2. Test a point in one of the half planes to
find whether it is a solution of the inequality.
77
We will test the point (0, 0) by substituting
into the original inequality.
False
So,we shade that side of the plane.
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Sketch the graph of
Example 2)
Graph the line
either solve for y, (slope-intercept form)
or find x- and y-int
Test an easy to deal with pointlike (0,0)
So, shade in the other side.
False
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Sketch the graph of
Example 3)
Graph the line
Test an easy to deal with pointlike (0,0)
So, shade in the other side.
False
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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
United States rate 0.16
(dollars per minute)
Labels
United States time x
(minutes)
Mexico rate 0.44
(dollars per minute)
(minutes)
Mexico time y
Value of card 50
(dollars)

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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.

0.16 x 0.44 y 50
Algebraic Model
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.
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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 point (0, 0). Because (0, 0) is a
solution if the inequality, shade the half-plane
below the line. Finally, because x and y cannot
be negative, restrict the graph to the points in
the first quadrant.
One solution is to spend 65 minutes on calls
within the United States and 90 minutes on calls
to Mexico.
Possible solutions are points within the shaded
region shown.
To split time evenly, you could spend 83 minutes
on calls 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.
84
Reflection on the Section
Describe the graph of a linear inequality in two
variables.
assignment
Page 111 1 - 44
2.6 Linear Inequalities in Two Variables
85
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
86
Goal
1
Represent piecewise functions
Piecewise functions are represented by a
combination of equations, each corresponding to a
part of the domain.
example 1)
Evaluate f(x) when (a) x -1, (b) x 1, and
(c) x 3
if x lt 0
if
if
2.7 Piecewise Functions
87
solution
Because -1 lt 0 , use first equation.
a. f(x) 2x 3 f(-1) 2(-1)3 1
Because , use second equation.
b. f(x) 2 f(1) 2
Because , use third equation.
c. f(x) -x1 f(3) -31 -2
2.7 Piecewise Functions
88
example 2)
Graphing a Piecewise Function
Graph the function
if
if
solution
To the left of x 3, the graph is
To the right of x 3, the graph is
2.7 Piecewise Functions
89
example 3)
Graphing a Piecewise Function
Graph the function
if
if
solution
To the right of x 1, the graph is
To the left of x 1, the graph is
2.7 Piecewise Functions
90
example 4)
Graphing a Step Function
Graph the function
if
if
if
solution
The graph is composed of three line segments,
because the function has three parts.
The intervals of x tell you that each line
segment is 2 units in length and begins with a
solid dot and ends with an open dot.
2.7 Piecewise Functions
91
example 4)
Graphing a Step Function
Graph the function
if
if
if
2.7 Piecewise Functions
92
Graph the function
2.7 Piecewise Functions
93
Graph the function
2.7 Piecewise Functions
94
Graph the function
2.7 Piecewise Functions
95
Reflection on the Section
A phone company charges in 60 second blocks.
What will a graph of the charges look like?
assignment
Page 117 13 - 30
2.7 Piecewise Functions
96
2.8
Absolute Value Functions
What you should learn
Goal
1
Represent and Graph absolute value functions.
Goal
2
Use Absolute Value functions to model real-life
situations.
2.8 Absolute Value Functions
97
Graphing Absolute Value Functions
The graph of has
the following characteristics.
The graph has vertex (h, k) and is symmetric in
the x h.
The graph is V-shaped. It opens up if a gt 0
and down if a lt 0.
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Graphing Absolute Value Functions
The graph of has
the following characteristics.
The graph has vertex (h, k) and is symmetric in
the x h.
The graph is V-shaped. It opens up if a gt 0 and
down if a lt 0.
The graph is wider than the graph
if
The graph is narrower than the graph
if
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x -3 -2 -1 0 1 2
3 2 1 0 1 2
100
But, before we get to graphing.
Solve the equation algebraically.
solution
or
The equation has two solutions 7 and -3. Check
these solutions by substituting each into the
original equation.
101
Check by Sketching the graph of the equation
rewrite
Find the coordinates of the Vertex.
x - 2 0 x 2
The coordinates for the vertex are (2,-5)
Now, make a table. Pick a couple of x-points less
than 2 and a couple of x-points greater than 2.
102
Solutions are the x-intercepts.
x 7 3 2 1 -3
0 -4 -5 -4 0
103
Graphs of Absolute Value Equations
How to graph an absolute value equation.
NO more graphing
104
Sorry
In this lesson we will learn to sketch the graph
of absolute value. To begin, lets look at the
graph of
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).
105
Vertex at (0,0) opens down
Vertex at (0,0) opens up
106
2
Vertex at (2,0), Opens up
Vertex at (0,1), Opens up
107
Sketching the graph of an Absolute Value
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 x-coordinate
of the vertex, some x-values to its left, and
some to its right.
3. Plot the points given, and connect.
108
Find the coordinates of the vertex of the graph
ex)
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)
109
Sketch the graph of the equation
ex)
Find the coordinates of the Vertex.
x 3 0 x -3
The coordinates for the vertex are (-3,-2)
Now, make a table. Pick a couple of x-points less
than -3 and a couple of x-points greater than -3.
110
x -5 -4 -3 -2 -1
0 -1 -2 -1 0
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Reflection on the Section
For the graph of Tell how to find the
vertex, the direction the graph opens, and the
slopes of the branches.
assignment
Page 125 1 - 25
2.8 Absolute Value Functions
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