college algebra chapter 2 functions and graphs. college algebra chapter 2.1 rectangular coordinates...

College AlgebraChapter 2

Functions and Graphs

College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line

Characteristics of a Linear Equation

1. Exponent on any variable is 12. No variable is used as a divisor3. No two variables are multiplied together


Linear Equations Terminology

Solution to a linear equation in 2 variables

Input / Output Table

Rectangular coordinate system

Origin

X axis

Y axis

Coordinate plane

Any pair of substitutions for x and y that result in a true equation

Way to organize solutions of linear equations

Consists of horizontal and vertical number lines

Where the horizontal and vertical number lines intersect

The horizontal number line

The vertical number line

Two dimensional plane where functions are graphed


Linear Equations Terminology

Quadrants

Gridlines / tick marks

Coordinate grid

Lattice points

Y-intercept

X-intercept

Begin with I in upper right and move counterclockwise

Placed on each axis to denote the integer values

When tick marks are extend throughout the coordinate plane

When both the x and y have integer values

Where the graph cuts through the x axis

Where the graph cuts through the y axis


Graph using the intercept method

2x+5y=6

3x-6y=18

-2x+y=-7


Graph the lines and tell where they intersect

x=4 y=-2

(4,-2)

Steepness of a line is referred to as slope

Measured using the ratio

The slope triangle

Slope expresses a rate of change between 2 quantities


Slope Formula and Rates of Change

changehorizontal

changevertical

"" inchangemeansx

y

xinchange

yinchange


12

12

xx

yy

run

rise

changehorizontal

changevertical

Slope Formula and Rates of Change

Result is called the slope formula


Use slope formula to calculate slope of lines that contain the following points

10,203,4 and

87,1229,9 and

3

58m

16

13m


Positive and Negative Slope

If m>0, then y values increase from left to right

If m<0, then y values decrease from left to right


Slope of horizontal and vertical lines

Horizontal line m=0

Vertical line m=undefined


Slope of parallel and perpendicular lines

Slope of parallel lines are equal

Slope of perpendicular lines are negative reciprocals

21

1

mm


Midpoint formula

Distance formula

2,

22121 yyxx

2122

12 yyxxd


2,

22121 yyxx

2122

12 yyxxd

Calculate midpoint and distance for each set of points

(-3,-2) and (5,4)

(7,10) and (-3,-10)


Homework pg 150 1-86

College Algebra Chapter 2.2 Relations, Functions, and Graphs

Relations and mapping notation

A relation is a correspondence between two sets


Relations and mapping notation

The set of all first coordinates is called the Domain

The set of all second coordinates is called the Range


Graph the following relation

xxy 22


Graph the following relation

29 xy


Graph the following relation2yx


A function is a relation where each element of the domain corresponds to exactly one element of the range

Vertical Line TestIf every vertical line intersects the graph of a relation in at most one point, the relation is a function

Domain uses vertical boundary lines

Range uses horizontal boundary lines


Domain of rational and square root functions

2

3

x

y

9

52

x

xy

32 xy

,22,x

3,3,| xRxx

,2

3x


Function Notation

542 2 xxxfGivenfind

2

2

2

3

2

af

af

f

f

College Algebra Chapter 2.3 Linear Functions and Rates of Change

Solving for y in linear equations ax+by=c offers advantages when evaluating

When a function has been solved for y (y has been written in terms of x) it is called function form


A linear function of the form y=mx+b, the slope of the line is m, and the y-intercept is

(0,b)

Solve each equation for y, then state the slope and y-intercept

3x-2y=18 4x-y=2


Finding equations when given the slope and a point

2,63

2m 8,10

2

1m 10,4

2

3m


Equations in point-slope form

mxx

yy

1

1Solve for y


Equations in point-slope form

11 yxxmy


Equations of lines parallel and perpendicular


Write the equations of the lines parallel and perpendicular to

4,625

4 throughxxf

College Algebra Chapter 2.4 Quadratic and other Toolbox Functions

Characteristics of Quadratics

Concavity

Axis of Symmetry

Vertex

Direction branches point, this is the end behavior (concave up or down)

Imaginary line that cuts the graph in half

Highest or lowest point


Graphing factorable quadratic functions

1. Determine end behavior: concave up if a > 0, concave down if a < 0

2. Find the y-intercept by substituting 0 for x:

3. Find the x-intercept(s) by substituting 0 for f(x) and solving for x

4. Find the axis of symmetry5. Find the vertex6. Use these features to help sketch a

parabolic graph

221 xx

h

khhfh ,,


Graphing factorable quadratic functions1. Determine end behavior:

concave up if a > 0, concave down if a < 0



4. Find the axis of symmetry5. Find the vertex6. Use these features to help

sketch a parabolic graph

221 xx

h

khhfh ,,

352 2 xxxfgraph


Graphing factorable quadratic functions1. Determine end behavior:

concave up if a > 0, concave down if a < 0



4. Find the axis of symmetry5. Find the vertex6. Use these features to help

sketch a parabolic graph

221 xx

h

khhfh ,,

1032 xxxfgraph


Square root functions

Always begin at a node

Also called a one wing graph

To graphFind the node, x-intercept, y-intercept, and an additional point



Square root functions 12 xxfgraph



Square root functions 2132 xxfgraph


Cubing Function

Have points of inflection / pivot points

To graph find1. End behavior2. Y-intercept3. x-intercepts4. Point of inflection5. Additional points

College Algebra Chapter 2.4 Quadratic and other Toolbox FunctionsTo graph find1. End behavior2. Y-intercept3. x-intercepts4. Point of inflection5. Additional points

Cubing Function xxxfgraph 43

College Algebra Chapter 2.4 Quadratic and other Toolbox FunctionsTo graph find1. End behavior2. Y-intercept3. x-intercepts4. Point of inflection5. Additional points

Cubing Function 4423 xxxxfgraph


Cube root function

Graph by selecting inputs that yield integer value outputs


3 3 xxfgraph

Cube root function


The average rate of change for a function

Given that f is continuous on the interval containing x1 and x2, the average rate of change of f between x1 and x2 is given by

12

12

xx

xfxf

x

y


The average rate of change for a function

12

12

xx

xfxf

x

y

Find the average rate of change for the interval 50 t

ttth 8016 2

College Algebra Chapter 2.5 Functions and Inequalities

Since an x-intercept is the input value that gives an output of zero, it is also referred to as a zero of a function


0,2

3

2

1 xginequalitythesolvexxgfor

Meaning for what inputs is the graph above or equal to the x axis?

What is the x-intercept?


0;542 xhxxxh

Solving quadratic inequalities

Find zeros of function

Check concavity

Sketch the parabola

State solution

College Algebra Chapter 2.6 Regression Technology and Data Analysis

Regression is an attempt to find an equation that will act as a model for raw data

College Algebra Chapter 2.6 Regression Technology and Data Anaylsis

Scatter Plots & positive / negative Association

Scatter Plots & linear / nonlinear associations

Strong & Weak Associations

Calculating linear equation model for a set of data

Linear regression and the line of best fit TI-83

College Algebra Chapter 2.6 Regression Technology and Data Anaylsis


College Algebra Chapter 2 Review

SlopeOrdered pairInterceptRangex-axisOutputFunctionDomainparallel linesRegressionperpendicular

linesscatter-plotaxis of symmetryOriginy-axisInputVertexlattice pointNodeRelationzeros of a function

College Algebra Chapter 2 Review

college algebra chapter 2 functions and graphs. college algebra chapter 2.1 rectangular coordinates...

Documents

line slide

y axis slide

rectangular coordinates

slope formula slide

graph cuts

parabolic graph

line slope formula

y values