Transcript B-5 Graphing inequalitiesx

Linear Relations and
Functions
B-5
Graphing Inequalities
Objectives
Graph linear inequalities
Graph absolute value inequalities
Essential Question
How do you determine which
region to shade when graphing
an inequality?
A linear inequality resembles a
linear equation, but with an
inequality symbol instead of an
equality symbol. For example,
y ≤ 2x + 1 is a linear inequality and
y = 2x + 1 is the related linear
equation.
The graph of y = 2x + 1 separates
the coordinate plane into two
regions. The line is the boundary
of each region. The graph of the
inequality y ≤ 2x + 1 is the shaded
region. Every point in the shaded
region satisfies the inequality. The
graph of y = 2x + 1is drawn as a
solid line to show that points on
the line satisfy the inequality. If
the inequality symbol were < or >,
then points on the boundary would
not satisfy the inequality, so the
boundary would be drawn as a
dashed line.
y = 2x + 1
y ≤ 2x + 1
You can graph an inequality by
following these steps:
Determine whether the boundary should
be solid or dashed. Graph the boundary.
Choose a point not on the boundary and
test it in the inequality.
If a true inequality results, shade the
region containing your test point. If a false
inequality results, shade the other region.
Graph
The boundary is the graph
of
Since the
inequality symbol is <, the
boundary will be dashed. The
x-intercept is (4, 0) and the yintercept is (0, −2).
Graph
Test (0, 0).
Original
inequality
true
Shade the region that contains (0, 0).
Graph
Answer:
Inequalities can sometimes be used to model real-world situations.
Education The SAT has two parts. One tutoring
company advertises that it specializes in helping
students who have a combined score on the SAT
that is 900 or less.
Write an inequality to describe the combined scores
of students who are prospective tutoring clients.
Let x be the first part of the SAT and let y be the
second part. Since the scores must be 900 or
less, use the  symbol.
The
1st part
x
Answer:
and
2nd part
together
are less than
or equal to
900.
y

900
Graph the inequality.
Since the inequality
symbol is , the graph
of the related linear
equation
is solid. This is the
boundary of the inequality.
Graph the inequality.
Test (0, 0).
Original
inequality
true
Graph the inequality.
Shade the region that
contains (0, 0). Since the
variables cannot be
negative, shade only the
part in the first quadrant.
Does a student with a verbal score of 480 and a math
score of 410 fit the tutoring company’s guidelines?
The point (480, 410) is in
the shaded region, so it
satisfies the inequality.
Answer: Yes, this student fits the tutoring
company’s guidelines.
Class Trip Two social studies classes are going
on a field trip. The teachers have asked for parent
volunteers to also go on the trip as chaperones.
However, there is only enough seating for 60 people
on the bus.
a. Write an inequality to describe the number of
students and chaperones that can ride on the bus.
Answer:
b. Graph the inequality.
Answer:
c. Can 45 students and 10 chaperones go on the trip?
Answer: yes
You can define an absolute value function as
𝑓 𝑥 = 𝑥 and is the parent function for the
family of all absolute value functions. The
graph of 𝑓 𝑥 = 𝑥 is V-shaped and is
symmetric about the y-axis. So, for every
point (x, y) on the graph, the point (−x, y) is
also on the graph. The
highest or lowest point on
the graph of an absolute
value function is called the
vertex. The vertex of the
parent function is (0, 0).
You can derive new absolute value
functions from the parent function through
transformations of the parent graph. A
transformation changes a graph’s size,
shape, position, or orientation. A translation
is a transformation that shifts a graph
horizontally and/or vertically, but does not
change its size, shape, or orientation.
The graph of 𝑦 = 𝑎 𝑥 − ℎ + 𝑘 is the graph of
𝑦 = 𝑥 translated h units horizontally and k
units vertically. The a represents a vertical
stretch if 𝑎 > 1 and is narrower. If 𝑎 < 1,
then the graph is compressed or wider than the
parent graph. The vertex of 𝑦 = 𝑥 − ℎ + 𝑘 is
(h, k). When a = −1, the
graph of 𝑦 = 𝑎 𝑥 is a
reflection across the x-axis.
Graph Absolute Value Inequalities
Graphing absolute value inequalities is
similar to graphing linear inequalities.
The inequality symbol determines
whether the boundary is solid or
dashed, and you can test a point to
determine which region to shade.
Graph
Since the inequality
symbol is , the graph
of the related equation
is solid and has
moved down 2 units from
the parent graph.
Graph the equation.
Test (0, 0).
Shade the region that
contains (0, 0).
Original inequality
true
Graph
Answer:
Essential Question
How do you determine which
region to shade when graphing
an inequality?
Choose a point not on the boundary and test
it in the inequality. If a true inequality results,
shade the region containing your test point. If
a false inequality results, shade the other
region. The point (0,0) is the easiest to use if
it is not on the boundary.
Math Humor
Teacher: Why are all your transformations
in French?
Student: They’re translations.