Transcript Intermediate Algebra 098A

Intermediate Algebra 098A
Chapter 8 and section 3.6
More on Functions and Graphs
Section 3.6
• Relations and Functions
• Review of Graphing
Calculator
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Jackie Joyner-Kersee - athlete
•“It is better to look
ahead and prepare
than to look back and
regret.”
Relation
• A set of Ordered Pairs.
• {1,2,(3,4)}
• {(2,3),(2,4)}
Domain
• The set of first components
of ordered pairs.
• {(1,2),(3,4)}
• Domain = {1,3}
Range
• The set of second
components of ordered pairs.
• {(1,2),(3,4)}
• Range = {2,4}
Function
• Is a relation in which no two
ordered pairs have the same
first components.
• {(1,2),(3,4)}
Vertical Line Test
• The graph of a relation
represents a function if and
only if no vertical line
intersects the graph at more
than one point
Interval Notation
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(2,5)
(2,5]
[2,5]
[2,5)
[2, )
(,2]
(2, )
(,2)
Section 3.6 continued
• Function Notation
• and
• Evaluation
Functional Notation
• f(x) read “f of x”
• Name of the function is f
• x is the domain element
• f(x) is the value of the range
Calculator evaluation
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Table
Y=
YVARS
Program Evaluate
Plug In
Store feature
Calculator Keys
• [VARS]
• [Y-VARS]
• Evaluating function
• Try Y = 3x – 2 for x =5
Repeated Evaluation of
expression
• Enter expression in [Y=]
• [VARS][Y-VARS]
• [1:Function]
• [1:Y][ENTER]
Evaluate Expression
• Enter expression in Y screen
• And produce table
nd
• 2 TBLSET
nd
• 2 TABLE
Graphing calculator keys
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[Y=]
[Window]
[Graph]
[Trace]
[Zoom]
[Zoom Integer]
Setting Window
• By Hand
• Zoom
• 6:Zstandard
• 8:Zinteger
• X[-9.4,9.4] Y[-10,10] friendly
window
• Zbox
• Zoom In
• Zoom Out
• Z Decimal
Modeling
• Algebraic Models of situations are not
perfect.
• Values of dates and variables need to be
examined carefully
• Models can give predictions
• Some models are better than others
Intermediate Algebra 3.6
•****Odell’s Calculator
Expectations
•Analysis of Functions
•Calculator Capabilities
Lou Holtz – football coach
•“No one has ever
drowned in sweat.”
Absolute Minimum
• Y-coordinate of the lowest
point of the graph of the
function.
Local Maximum
• Highest point in a
“neighborhood”
• Local Minimum
• Lowest point in a
“neighborhood.”
Points of Intersection
• The point(s) at which the two
graphs of two function on the
same set of axes intersect each
other.
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Intermediate Algebra 098A 8.1
Graphing and Writing Linear Equations
Review of equations of Lines
Use of Graphing Calculator
Calculator Keys
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2nd CALC
ZERO
MINIMUM
MAXIMUM
INTERSECT
VALUE
• Study Groups are Useful
Unknown author
• “Today, be aware of how you
are spending your 1,440
beautiful moments, and spend
them wisely.”
Intermediate Algebra 098A
Chapter 8.1
• Graphing and
• Writing Linear Functions
Linear functions
• Properties
of
Lines
- Review
Def: Linear Equation
• A linear equation in two
variables is an equation that
can be written in standard
form ax + by = c where a,b,c
are real numbers and a and b
are not both zero
Def: Intercepts
• y-intercept – a point where a
graph intersects the y-axis.
• x-intercept is a point where a
graph intersects the x-axis.
Procedure to find intercepts
• To find x-intercept
• 1. Replace y with 0 in the given
equation.
• 2. Solve for x
• To find y-intercept
• 1. Replace x with 0 in the given
equation.
• 2. Solve for y
Horizontal Line
•y = constant
• Example: y = 4
• y-intercept (0,4)
• Function – no x intercept
Vertical Line
• x = constant
• Example x = -5
• x-intercept (-5,0)
• No y intercept
• Not a function
Slope
rise y2  y1
y
m


run x2  x1
x
Horizontal line
• y = constant
• Slope is 0
• Examples: y = 5
• y = -3
• Can be done with calculator.
Vertical Line
• x=constant
• Undefined slope
• Examples:
• x =2
• x = -3
• Not graphed by calculator
Slope Intercept Form for
equation of Line
• y=mx+b
Slope is m
y-intercept is (0,b)
Using Slope Intercept form to
graph a line
• 1. Write the equation in form y=mx+b
• 2. Plot y intercept (0,b)
• 3. Write slope with numerator as positive
or negative
• 3. Use slope – move up or down from y
intercept and then right- plot point.
• 4. Draw line through two points.
Problem
• The percentage B of automobiles
with airbags can be modeled by the
linear function B(t)-5.6t –3.6,
where t is the number of years
since 1990.
• What is the slope of the graph of
B?
• Answer is 5.6
Fred Couples – Professional
Golfer
• “When you’re prepared
you’re more confident:
when you have a strategy
you’re more
comfortable.”
Objectives:
• Determine if two lines are
parallel.
• Determine if two lines are
perpendicular.
Def: Parallel Lines
• Two distinct non-vertical
lines are parallel if and only
if they have the same slope.
• Two distinct vertical lines are
parallel.
Def 2: Perpendicular Lines
• The slope of a line
perpendicular to another line is
the negative reciprocal of the
slope of the original line.
• If slope is a/b, slope of
perpendicular line is –b/a.
Helen Keller – advocate for he
blind
• “Alone we can do so
little, together we can
do so much.”
Objective
• Use slope-intercept
form to write the
equation of a line.
y=mx+b
• Write the equation of a line
given the slope and the y
intercept.
• Line slope is 2 and y intercept
(0,-3)
• y=2x-3
y=mx+b
• Write the equation of a line
given the slope and one
point.
• Slope of 2 and point (1,3)
• y=2x+1
Point-slope form
of Linear equation
given slope of m & pt ( x1 , y1 )
y  y1  m( x  x1 )
Objective: Write equation of a
line given the slope and one point
• Problem: slope of –3 through
(2,-4)
• Answer: y=-3x+2
Objective – Write equation of
line given two points
• Given points (-3,6),and (9,-2)
• Find slope
• Slope is –2/3
• Answer: y=(-2/3)x+4
Objective: Write equation of a
line in slope-intercept form that
passes through (4,-1) and is
parallel to y=(-1/2)x+3
• y=(-1/2)x+1
Intercepts Form for equation of a
line.
• a is the x-intercept
• b is the y-intercept
x y
 1
a b
Pop Warner – football coach
•“You play the way
you practice.”
Section 8.4 – Gay
Variation and Problem Solving
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Direct Variation
Inverse Variation
Joint Variation
Applications
Def: Direct Variation
• The value of y varies
directly with the value of
x if there is a constant k
such that y = kx.
Objective
• Solve Direct Variation
Problems
• Determine constant of
proportionality.
Procedure:Solving Variation Problems
• 1. Write the equation
• Example y = kx
• 2. Substitute the initial values and
find k.
• 3. Substitute for k in the original
equation
• 4. Solve for unknown using new
equation.
Example: Direct Variation
• y varies directly as x. If y =
18 when x = 5, find y when x
=8
• Answer: y = 28.8
Helen Keller – advocate for he
blind
• “Alone we can do so
little, together we can
do so much.”
Definition: Inverse Variation
• A quantity y varies inversely with x if there
is a constant k such that
k
y
x
• y is inversely proportional to x.
• k is called the constant of variation.
Procedure: Solving inverse
variation problems
• 1. Write the equation
• 2. Substitute the initial values
and find k
• 3. Substitute for k in the
equation found in step 1.
• 4. Solve for the unknown.
Joint Variation
• Three variables y,x,z are
in joint variation if y =
kxz where k is a constant.
Leonardo Da Vinci - scientist,
inventor, and artist
• “Time stays long
enough for those who
use it.”