Intermediate Algebra 098A

Chapter 8 and section 3.6

More on Functions and Graphs

Section 3.6

• Relations and Functions

• Review of Graphing Calculator

• ***********************

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

• (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

• 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

• 2nd TBLSET

• 2nd TABLE

Graphing calculator keys

• [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.

• Intermediate Algebra 098A 8.1• Graphing and Writing Linear Equations• Review of equations of Lines• Use of Graphing Calculator

Calculator Keys

• 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 098AChapter 8.1

• Graphing and

• Writing Linear Functions

Linear functions

• PropertiesofLines- 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

2 1

2 1

rise y y ym

run x x 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

1 1& ( , )given slope of m pt x y

1 1( )y y m x x

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

1x y

a b

Pop Warner – football coach

•“You play the way you practice.”

Section 8.4 – GayVariation and Problem Solving

• 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

• y is inversely proportional to x.

• k is called the constant of variation.

ky

x

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.”