Transcript CHAPTER 2: LINEAR RELATIONS & FUNCTIONS

Chapter 2
Linear Relations
and
Functions
BY:
FRANKLIN KILBURN
HONORS ALGEBRA 2
Summary Slide
•2–1
Relations & Functions
• 2 – 1 Cont'd
• 2 – 1 Cont'd
•2–2
LINEAR EQUATIONS
• 2 – 2 Cont'd
Summary Slide (cont.)
•2–3
SLOPE
• 2 – 3 Cont'd
• 2 – 3 Cont'd
•2–4
WRITING LINEAR EQUATIONS
• 2 – 4 Cont'd
Summary Slide (cont.)
•2–5
Modeling Real-World Data:
Using Scatter Plots
• 2 – 5 Cont'd
•2–6
SPECIAL FUNCTION
• 2 – 6 Cont'd
Summary Slide (cont.)
•2–7
GRAPHING INEQUALITIES
• 2 – 7 Cont'd
• 2 – 7 Cont'd
• Examples of Boundaries
2–1
Relations & Functions
 Ordered pairs can
be graphed on a
coordinate system.
The Cartesian
coordinate plane is
composed of the xaxis (horizontal) and
the y-axis (vertical),
which met at the
origin (0,0) and
divide the plane into
four quadrants.
2 – 1 Cont'd

A relation is a set of
ordered pairs.

The domain of a relation
is the set of all first
coordinates (xcoordinates) from all the
ordered pairs, and the
range is the set of all
ordered coordinates from
all second coordinates (ycoordinates).


The graph of a relation is
the set of points in the
coordinate plane
corresponding to the
ordered pairs in the
relation.
A function is a special
type of relation in which
each element of the
domain is paired with
exactly one element of
the range.
 A mapping shows how each member of the domain
is paired with each member of the range
2 – 1 Cont'd
 A function where each
element of the range is
paired exactly one
element of the domain
is called a one-to-one
function.
• Vertical line test: if no
vertical line intersects
a graph in more than
one point, then the
graph represents a
function
• When an equation
represents a function
there are two sets of
variables:
• The independent
variable is usually x,
and the values make
up the domain.
• A dependent variable
usually y, has values
which depend on x.
 The equations are often written in functional notation.
Ex: y=2x+1 can be written as f(x)=2x+1. The symbol f(x)
replaces the y and is read “f of x”.
2–2
LINEAR EQUATIONS
• A linear equation has no operations
other than addition, subtraction, and
multiplication of a variable by a
constant.
• The variables may not be multiplied
together or appear in a denominator.
• Does not contain variables with
exponents other than 1.
• The graph is always a line.
2 – 2 Cont'd
 A linear function is a
function whose ordered
pairs satisfy a linear
equation. Any linear
function can be written
in the form f(x) = mx+b,
where m and b are real
numbers.
 Any linear equation
can be written in
standard form
– Ax+By=C –
where A, B, and C
are real numbers.
• The y-intercept is
the point of the
graph in which the
y-coordinate
crosses the y-axis.
 The x-intercept is
the point of the
graph in which the
x-coordinate
crosses the x-axis.
2–3
SLOPE
 The slope of a line is the ratio of
the changes in y-coordinates to
the change in x-coordinates. Slope
measures how steep a line is.
 A family of graphs is the group of
graphs that displays one or more
similar characteristics.
• The parent graph is the simplest
of the graphs in a family
2 – 3 Cont'd
• The rate
of change
measures
how much
a quantity
changes on
average,
relative to
the change
in another
quantity.
• The slope of a line
tells the direction in
which it rises of falls:
• If the line rises to the
right, the slope is
positive.
• If the line is horizontal,
the slope is zero.
• If the line falls to the
right, the slope is
negative.
• If the line is vertical, the
line is undefined.
2 – 3 Cont'd
• In a plane, non-vertical lines
with the same slope are
parallel. All vertical lines are
parallel.
• In a plane, two oblique lines are
perpendicular if and only if the
product of their slopes is -1.
2–4
WRITING LINEAR EQUATIONS
 Slope –
intercept
form is the
equation of
a line in the
form y=mx+b,
where m is
the slope
and b is the
y - intercept.
• An equation in the form
y = 4/3 x - 7
is the point slope form.
 The slope-intercept and pointslope forms can be said to find
equations of lines that are
parallel or perpendicular to
given lines.
2 – 4 Cont'd
• The point - slope form of the
equation of a line is y-y^1=m(xx^1) where (x^1,y^1) are the
coordinates of a point on the
line and m is the slope of the
line.
2–5
Modeling Real-World Data:
Using Scatter Plots
• Data with two
variables such as
speed and Calories is
called bivariate data.
• A set of bivarate
date graphed as
ordered pairs in a
coordinate plane is
called a scatter plot.
• A scatter plot can
show whether there
is a relationship
between the data.
2 – 5 Cont'd
• A scatter plot is a set of data
graphed as ordered pairs in a
coordinate plane.
• An equation suggested by the
points of a scatter plot used to
predict other points is called a
prediction equation.
• Line of fit: line that closely
approximates a set of data
2–6
SPECIAL FUNCTION
• A step function is a
function whose graph
is a series of line
segments.
• A greatest integer
function is a step
function, written as
f(x)=[[x]], where f(x)
is the greatest
integer less than or
equal to x.
• A constant function
is a linear function in
the form of f(x)=b.
2 – 6 Cont'd
• Identity function: the function of
1(x)=x
• A piecewise function is written
using two or more expressions
• A constant function is a linear
function in the form of f(x)=b.
2–7
GRAPHING INEQUALITIES
• A linear inequality resembles
a linear equation, but with an
inequality symbol rather than
an equal symbol. Ex: y<2x+1
is a linear inequality and
y=2x+1 is the related linear
equation.
2 – 7 Cont'd
• A boundary is a region bounded when
the graph of a system of constraints
is a polygonal region.
• Graphing absolute value inequalities
is similar to graphing linear equations.
The inequality symbol determines
whether the boundary is solid or
dashed, and you can test a point to
determine which region to shade.
2 – 7 Cont'd
• A linear
inequality
resembles a
linear equation,
but with an
inequality
symbol rather
than an equal
symbol.
Ex: y<2x+1 is a
linear inequality
and y=2x+1 is
the related
linear equation.
Examples of Boundaries
• Example 1
Dashed Boundary
• Example 2
Solid Boundary