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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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Graph using the intercept method
2x+5y=6
3x-6y=18
-2x+y=-7
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Slope Formula and Rates of Change
changehorizontal
changevertical
"" inchangemeansx
y
xinchange
yinchange
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
12
12
xx
yy
run
rise
changehorizontal
changevertical
Slope Formula and Rates of Change
Result is called the slope formula
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Slope of horizontal and vertical lines
Horizontal line m=0
Vertical line m=undefined
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Slope of parallel and perpendicular lines
Slope of parallel lines are equal
Slope of perpendicular lines are negative reciprocals
21
1
mm
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Midpoint formula
Distance formula
2,
22121 yyxx
2122
12 yyxxd
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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)
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
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
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Relations and mapping notation
The set of all first coordinates is called the Domain
The set of all second coordinates is called the Range
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Graph the following relation
xxy 22
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Graph the following relation
29 xy
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Graph the following relation2yx
College Algebra Chapter 2.2 Relations, Functions, and Graphs
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
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Domain of rational and square root functions
2
3
x
y
9
52
x
xy
32 xy
,22,x
3,3,| xRxx
,2
3x
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Function Notation
542 2 xxxfGivenfind
2
2
2
3
2
af
af
f
f
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Homework pg 167 1-110
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
College Algebra Chapter 2.3 Linear Functions and Rates of Change
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
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Finding equations when given the slope and a point
2,63
2m 8,10
2
1m 10,4
2
3m
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Equations in point-slope form
mxx
yy
1
1Solve for y
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Equations in point-slope form
11 yxxmy
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Equations of lines parallel and perpendicular
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Write the equations of the lines parallel and perpendicular to
4,625
4 throughxxf
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Homework pg 182 1-114
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
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
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 ,,
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Graphing factorable quadratic functions1. 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 ,,
352 2 xxxfgraph
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Graphing factorable quadratic functions1. 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 ,,
1032 xxxfgraph
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
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
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
To graphFind the node, x-intercept, y-intercept, and an additional point
Square root functions 12 xxfgraph
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
To graphFind the node, x-intercept, y-intercept, and an additional point
Square root functions 2132 xxfgraph
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
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
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Cube root function
Graph by selecting inputs that yield integer value outputs
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
3 3 xxfgraph
Cube root function
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
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
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
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 2.4 Quadratic and other Toolbox Functions
Homework pg 200 1-84
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
College Algebra Chapter 2.5 Functions and Inequalities
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?
College Algebra Chapter 2.5 Functions and Inequalities
0;542 xhxxxh
Solving quadratic inequalities
Find zeros of function
Check concavity
Sketch the parabola
State solution
College Algebra Chapter 2.5 Functions and Inequalities
Homework pg 212 1-92
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
Homework pg 224 1-38
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 Review
College Algebra Chapter 2 Review
College Algebra Chapter 2 Review