College Algebra
Chapter 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 1
2. No variable is used as a divisor
3. 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
Any pair of substitutions for x and y that result in a true equation
Input / Output Table
Way to organize solutions of linear equations
Rectangular coordinate system
Consists of horizontal and vertical number lines
Origin
X axis
Where the horizontal and vertical number lines intersect
The horizontal number line
Y axis
The vertical number line
Coordinate plane
Two dimensional plane where functions are graphed
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Linear Equations Terminology
Quadrants
Begin with I in upper right and move counterclockwise
Gridlines / tick marks
Placed on each axis to denote the integer values
Coordinate grid
When tick marks are extend throughout the coordinate plane
Lattice points
When both the x and y have integer values
Y-intercept
Where the graph cuts through the x axis
X-intercept
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)
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Slope Formula and Rates of Change
Steepness of a line is referred to as slope
vertical change
Measured using the ratio
horizontal change
The slope triangle
Slope expresses a rate of change between 2 quantities
changein y y

changein x x
 m eans" changein"
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Slope Formula and Rates of Change
vertical change
rise
y2  y1


horizontalchange run
x2  x1
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
9,29 and 12,87
m
58
3
4,3 and 20,10
m
13
16
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
1
m1  
m2
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Midpoint formula
 x1  x2 y1  y2 
,


2 
 2
Distance formula
d
x2  x1    y2  y1 
2
2
College Algebra Chapter 2.1 Rectangular Coordinates and the Graph of a Line
Calculate midpoint and distance for each set of points
d
(-3,-2) and (5,4)
(7,10) and (-3,-10)
x2  x1    y2  y1 
2
2
 x1  x2 y1  y2 
,


2 
 2
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
y  x  2x
2
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Graph the following relation
y  9 x
2
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Graph the following relation
2
x y
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 Test
If 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
3
y
x2
x   ,2   2, 
x5
y 2
x 9
x | x  R, x  3,3
y  2x  3
 3 
x  ,
 2 
College Algebra Chapter 2.2 Relations, Functions, and Graphs
Function Notation
Given f x  2x  4x  5
2
find
f  2 
3
f 
2
f 2a 
f a  2 
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
m
3
6,2
1
m
2
 10,8
3
m
2
4,10 
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Equations in point-slope form
y  y1
m
x  x1
Solve for y
College Algebra Chapter 2.3 Linear Functions and Rates of Change
Equations in point-slope form
y  mx  x1   y1
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
f x   x  2 through
5
 6,4
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
Direction branches point, this is the end behavior
(concave up or down)
Axis of Symmetry
Imaginary line that cuts the graph in half
Vertex
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
x x
4. Find the axis of symmetry h  2
5. Find the vertex h, f h  h, k 
6. Use these features to help sketch a parabolic
graph
1
2
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Graphing factorable quadratic functions
graph f x  2x  5x  3
2
1.
2.
3.
4.
5.
6.
Determine end behavior: concave
up if a > 0, concave down if a < 0
Find the y-intercept by substituting
0 for x:
Find the x-intercept(s) by
substituting 0 for f(x) and solving
for x
x x
Find the axis of symmetry h  1 2
Find the vertex h, f h  h, k  2
Use these features to help sketch a
parabolic graph
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Graphing factorable quadratic functions
graph f x  x  3x 10
2
1.
2.
3.
4.
5.
6.
Determine end behavior: concave
up if a > 0, concave down if a < 0
Find the y-intercept by substituting
0 for x:
Find the x-intercept(s) by
substituting 0 for f(x) and solving
for x
x x
Find the axis of symmetry h  1 2
Find the vertex h, f h  h, k  2
Use these features to help sketch a
parabolic graph
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 graph
Find the node, x-intercept, y-intercept, and an additional point
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Square root functions
graph f x  x  2 1
To graph
Find the node, x-intercept,
y-intercept, and an
additional point
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Square root functions
graph f x   2x  13  2
To graph
Find the node, x-intercept,
y-intercept, and an
additional point
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Cubing Function
Have points of inflection / pivot points
To
1.
2.
3.
4.
5.
graph find
End behavior
Y-intercept
x-intercepts
Point of inflection
Additional points
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Cubing Function
graph f x  x3  4 x
To graph find
1. End behavior
2. Y-intercept
3. x-intercepts
4. Point of inflection
5. Additional points
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
Cubing Function
graph f x  x3  x 2  4x  4
To graph find
1. End behavior
2. Y-intercept
3. x-intercepts
4. Point of inflection
5. Additional points
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
Cube root function
graph f x  3 x  3
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
y f x2   f x1 

x
x2  x1
College Algebra Chapter 2.4 Quadratic and other Toolbox Functions
The average rate of change for a function
Find the average rate of change
for the interval 0  t  5
ht   16t 2  80t
y f x2   f x1 

x
x2  x1
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
1
3
for g x    x  , solve the inequality g x   0
2
2
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
Solving quadratic inequalities
hx  x 2  4x  5; hx  0
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
lines
scatter-plot
axis of symmetry
Origin
y-axis
Input
Vertex
lattice point
Node
Relation
zeros of a function
Slope
Ordered pair
Intercept
Range
x-axis
Output
Function
Domain
parallel lines
Regression
perpendicular
College Algebra Chapter 2 Review
College Algebra Chapter 2 Review
College Algebra Chapter 2 Review
College Algebra Chapter 2 Review