BY:
Jered Johnson
HONORS ALGEBRA 2
2–1
Relations & Functions
 2 – 1 Cont'd
 2 – 1 Cont'd
2–2
LINEAR EQUATIONS
 2 – 2 Cont'd
2–3
SLOPE
 2 – 3 Cont'd
 2 – 3 Cont'd
2–4
WRITING LINEAR EQUATIONS
 2 – 4 Cont'd
2–5
Modeling Real-World Data: Using Scatter Plots
 2 – 5 Cont'd
2–6
SPECIAL FUNCTION
 2 – 6 Cont'd
2–7
GRAPHING INEQUALITIES
 2 – 7 Cont'd
 2 – 7 Cont'd
 Examples of Boundaries
2–1
 Ordered pairs can be
graphed on a coordinate
system. The Cartesian
coordinate plane is
composed of the x-axis
(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.
 The y-intercept is the
point of the graph in
which the y-coordinate
crosses the y-axis.
 Any linear equation can be
written in standard form
– Ax+By=C –
where A, B, and C
are real numbers.
 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 point-slope
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(x-x^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.
Examples of Boundaries
 Example 1
Dashed Boundary
 Example 2
Solid Boundary