LL
1
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Section 2.1
Relation:
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Relations and Functions
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SQl:
Example 1: Graphing a Relation
Graph the relation {(-2, 4), (3, -2), (-1, 0), (1, 5)}.
Domain: fry
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Range:
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cHQ
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Example 2: Finding Domain and Range
Write the ordered pairs for the relation shown in the graph. Find the domain and range.
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Mapping Diagram: flQth[ \)3
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Example 3: Making a Mapping Diagram
Make a mapping diagram for the relation {(-1,-2), (3, 6), (-5, -10), (3, 2)}.
Function:
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np;J
Yp±
r’i
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fcc
Example 4: Identifying Functions
Determine whether each relation is a function.
b.
a.
Doinan
Range
Range
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c 3.
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R -Rnc+1crL SncQ QCO np*
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Vertical-Line Test: 1J-c
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1fr cook
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Example 5: Usmg the Vertical-Lme Test
Use the vertical-line test to determine whether each graph represents a function.
b.
a.
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Function Rule:
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Function Notation:
Example 6:
Findf(-3),J(O), andf(5) for each function.
a. f(x)=3x—5
3
Ei
=fEI1
=
3
b. f(a)=—a—1
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c. f(y)=——y+-
=
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Linear Equations
Section 2.2
Linear Function:
1
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Linear Equation:
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Dependent Variable:
Independent Variable: —-\.
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Example 1: Graphing a Linear Equation
Graph the equation y
2
=
—
x+3
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—
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SIope:_,
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Slope Formula:
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ch Cc
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Example 2: Finding Slope
V.
Find the slope of the line through points (3, 2) and (-9, 6.
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(1-
0)
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Example 5: Graphing Lines Using Intercepts
Graph 6x+3y=12.
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Exampl 6:
The equation 3x+ 2y = 120 models the number of passengers who can sit in a train car,
where x is the number of adults and y is the number of children. Graph the equation.
Explain what the x- and y-intercepts represent. Describe the domain and range.
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Example 7: Transforming Into Slope-Intercept Form
Graph 4x—2y=9.
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4
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=Zx
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Point-Slope Form:
The line through point 1
(x 1
,
y with slope
)
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has the equation:
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Example 8: Writing an Equation Given the Slope and a Point
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Write in standard form an equation of the line with slope
through the point (8, -1).
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+O (x-)
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+
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Example 9: Writing an Equation Given Two Points
Write in point-slope form the equation of the line through (1, 5) and (4, -1).
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Summary of Equations of a Line:
Point-Slope Form
Standard Form
7
Slope-Intercept Form
Lines with slopes that have special properties:
Horizontal
Vertical Line
tine
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Parallel Lines
Perpendicular Lines
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+
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Example 10: Writing Equations of Parallel Lines
Write an equation of a line parallel to y = —4x+ 3 that contains the point (1, -2).
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Example 11: Writing an Equation of a Perpendicular Line
x +2. Graph
Write an equation of the line through each point and perpendicular to y
all three lines.
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ppenthcoir
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a. (0, 4)
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b. (6, 1)
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2.4 — Using Linear Models
Example 1:
Jacksonville, Florida has an elevation of 12 ft above sea level. A hot-air balloon taking
off from Jacksonville rises 50 flJmin.
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a. Write an equation to model the balloon’s
elevation as a function of time.
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b. Graph the equation.
c. Interpret the intercept at which the graph
intersects the vertical axis.
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Example 2:
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A candle is 6 in. tall after burning for 1 h. After 3 h, it is 5— in. tall. Write a linear
equation to model the height y of the candle after burning x hours.
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after 3 h
Example 3: Using a Linear Model
Use the equation from example 2. In how many hois will the candle be 4 in. tall?
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Scatter Plot:
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Trend Line:
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Example 4:
A woman is considering buying a 1999 car for $4200. She researches prices for various
years of the same model and records the data in a table.
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Modet Year
2000
2001
20O2.200.,. 2004
Prices.
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$8. $9660
$io,94s
$5435
$62f 17
$7751
$9127
$1ft455
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a. Let x represent the model year. (Use 1 for 2000, 2 for 2001, and so forth.) Lety be
the price of the car. Draw a scatter plot and decide whether a linear model is
reasonable.
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b. Draw a trend line. Write the equation of the
line. Determine whether the asking price
is reasonable.
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Absolute Value Functions and Graphs
Section 2.5
n
Absolute Value Function:
Vertex:
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Example 1: Graphing an Absolute Value Function
a. Graph y = 3x +12j.
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Example 2: Using a Graphing Calculator
+6.
Graph
y=—13x+41
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12
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Families of Functions
Section 2.6
Parent Function:
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Translation:
c
Vertical Translation:
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±
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Horizontal Translation:
coc
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Example 1: Vertical Translation
a. Describe the translation y
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Ix —3 and draw its graph.
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b. Write an equation to translate y =
-kçcii C\ y\
unit.
up
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Example 2: Horizontal Translations
a. The graph with the vertex at x
graph.
b. Describe the translation
tcJcDy\
cc
=
5 is a translation of y = x. Write an equation for the
+3
and draw its graph.
x( o
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Summary of The Family of Absolute Value Functions:
Vertical Translation
Parent Function:
I I
Translation up k units, k> 0:
x( 4
=
\(
Translation down k units, k> 0:
Ix H
Horizontal Translation
Parent Function:
I
\)
Translation right h units, h> 0:
X
Translation left h units, h> 0:
Combined Translation
(right h units, up k units)
IX
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t
j + ft
Vertical Stretch:
,
(Qc
1> S**c
b c
ccph
Vertical Shrink:
0
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Example 3: Graphing y
ccu
lxi.
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b. Write an equation for a vertical shrink of y =
Reflection:
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a. Describe and then draw the graph of y =2
(1
fcc-ci
lxi by a factor of
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Example 4: Graphing y = —a
I.
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lxi
Which equation describes the graph?
a.
b.
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1
1
y=—x
I
1
(c.’ y=——x
2
1
d. y=—.-x
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______
______
__________
Two Variable Inequalities
Section 2.7
Linear Inequality:
c)-
pi
;
_-
QoccoQ
For an inequality with
For an inequality with
>
bondd
1 c v’ Q
b\
QOk ‘
or
shade below the line.
or
shade above the line.
1;+
v
A solid boundary line indicates that the
line is part of the solution.
y>
—
A dashed boundary
line indicates that
the line is not part
of the solution.
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I
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Example 1: Graphing a Linear Inequality
x —3.
a. Graph the inequality y
Step 1: Graph the boundary line.
Step 2: Shade the appropriate region.
(c,-’)
Cc:
Lc3
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b. Graph the inequality 3x 4y
—
12
...
..
Step 1: Solve for y
.
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Ni
.-q
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—
Step 2: Graph the boundary line.
N
Step 3: Shade the appropriate region.
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Example 2: Graphing Absolute Value Inequalities
Graph each absolute value inequality.
a. yjx—4+5
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7
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k.
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b. —y+3>x+l
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— —
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Example 3: Writing Inequalities
The graph is the solution of which inequality?
a. y>x—3I+2
b. y<jx—31+2
) yx—3+2
d. y)x—3J+2
-
z
-